The action aims to diplomatically isolate Russia at the world body.
The resolution, supported by 141 of the assembly's 193 members, ended a rare emergency session called by the UN Security Council and as Ukrainian forces battled on in the port of Kherson in the face of air strikes and a devastating bombardment that forced hundreds of thousands of people to flee.
The text of the resolution "deplores" Russia's "aggression against Ukraine”.
Thirty-five members including China abstained and five countries including Russia, Syria and Belarus voted against the resolution.
While General Assembly resolutions are non-binding, they carry political weight.
Ukrainian President Volodymyr Zelensky wrote on Twitter that the countries that backed the vote “have chosen the right side of history”.
The resolution came as Russia renewed its aerial assault on Ukraine’s second-largest city in a pounding that lit up the skyline with balls of fire over populated areas.
The sustained assault came as even as both sides said they were ready to resume talks aimed at stopping the fighting.
Seven days into the invasion, a refugee crisis is unfolding on the European continent, with the United Nations saying that more than 870,000 people have fled Ukraine and that the number could soon hit one million.
Ukraine’s State Emergency Service reported that more than 2,000 civilians have been killed, but that could not immediately be independently verified, and neither side has disclosed its military casualties.
Kharkiv, Ukraine's second-biggest city, with a population of about 1.5m, came under bombardment again today, and a strike reportedly hit a hospital in the country’s north.
Meanwhile, a 64km column of Russian tanks and other vehicles stood outside the capital, Kyiv, while invading forces pressed their assault on the strategic port cities of Kherson and Mariupol in the south.
The two sides held talks on Monday, agreeing only to keep talking.
It was not immediately clear when new talks might take place or what they would yield.
Mr Zelensky said on Tuesday that Russia should stop bombing before another meeting.
The president has decried Russia’s attacks on civilian targets as a blatant terror campaign, while US President Joe Biden warned on Tuesday that if Russian leader Vladimir Putin is not made to “pay a price” for the invasion, the aggression won't stop with one country.
Russia, too, ramped up its rhetoric, with Foreign Minister Sergei Lavrov reminding the world about the country's vast nuclear arsenal.
In an interview with Al-Jazeera, he said: “A third world war will be nuclear, and devastating," according to Russian news sites.
A Russian attack hit the regional police and intelligence headquarters in Kharkiv, killing four people and wounding several others, Ukraine's emergency service said.
It added that residential buildings were also hit, but did not provide details.
A blast blew the roof off of the five-story police building and set the top floor on fire, according to videos and photos released by the service. Pieces of the building were strewn across the streets.
In the northern city of Chernihiv, two cruise missiles hit a hospital, according to the Ukrainian UNIAN news agency, which quoted the health administration chief, Serhiy Pivovar, as saying authorities were working to determine the casualty toll.
In besieged Mariupol, at least one teenager died and two more were wounded by apparent Russian shelling.
The three boys were rushed to a regional hospital.
One had lost his legs in the attack and died soon after arriving, according to an Associated Press reporter at the scene.
Family members told the AP the three had been playing soccer near a school when the shelling hit.
The attacks came a day after Russia, intensifying its attacks on cities, bombed Kharkiv's central square — where at least six people were reported killed — and struck Kyiv's main TV tower, where authorities said five died.
Kyiv's nearby Babi Yar Holocaust memorial also came under fire, but the main monument was not damaged.
Zelensky expressed outrage at the attack near Babi Yar, where Nazi occupiers killed more than 33,000 Jews over two days in 1941.
Even as Russia pressed its assault, Kremlin spokesman Dmitry Peskov said that a delegation would be ready later in the day to meet Ukrainian officials.
Ukrainian Foreign Minister Dmytro Kuleba also said his country was ready but noted that Russia's demands have not changed and that he wouldn't accept any ultimatums. Neither side said where the talks might take place.
Meanwhile, jailed Russian opposition leader Alexei Navalny has shared a series of tweets calling for daily demonstrations across Russia and beyond.
Navalny has spent a year behind bars after surviving a poison attack that he blames on the Kremlin.
"We - Russia - want to be a nation of peace. Alas, few people would call us that now," he tweeted from the maximum-security prison east of Moscow where he is imprisoned.
But Russians should "at least not become a nation of frightened silent people" who "pretend not to notice the war".
"It's the third decade of the 21st Century, and we are watching news about people burning down in tanks and bombed houses.
"We are watching real threats to start a nuclear war on our TVs."
Day 7, today, of the biggest ground war in Europe since World War II found Russia increasingly isolated, beset by tough sanctions that have thrown its economy into turmoil and left the country practically friendless, apart from a few nations such as China, Belarus and North Korea.
Many military experts worry that Russia may be shifting tactics. Moscow’s strategy in Chechnya and Syria was to use artillery and air bombardments to pulverise cities and crush fighters’ resolve.
Overall death tolls from the fighting remained unclear, but a senior Western intelligence official estimated more than 5,000 Russian soldiers have been captured or killed. Ukraine has given no overall estimate of troop losses.
The United Nations says at least thirteen children are believed to be among the dead since the Russian invasion of Ukraine began last Thursday.
But Liz Throssell, a spokesperson for the UN High Commissioner for Human Rights, said the real death toll is likely to be much higher.
Meanwhile, the Russian invasion of Ukraine is creating a “real danger of a nuclear accident”, environmental campaigners have warned.
They said the country’s 15 reactors were vulnerable to accidental air strikes, loss of electricity for control operations and lack of access for monitoring staff.
Russian troops have taken over the Chernobyl plant in the north of the country.
While the damaged reactor from the 1986 disaster is safely contained, radioactive spent fuel from decades of power generation is in dilapidated facilities awaiting transfer to a newly built storage unit.
UN says Ukraine refugee surge will soon hit one millionUkraine and Russia: What you need to know today
Olexiy Pasyiuk, deputy director of Ecoaction Ukraine, said staff in Chernobyl, who are normally brought to and from the plant each day, were not allowed to rotate shifts and had their mobile phones taken.
Workers in the wider exclusion zone, who normally rotate fortnightly, had been evacuated, so routine monitoring was not being carried out.
A spike in radioactivity was recorded last week which some attributed to dust being unsettled by the sudden arrival of many trucks, but Mr Pasyiuk said he was not reassured.
“There are inconsistences. We don’t know how it got into the air when it was wet and trucks were getting stuck in the mud.”
The State Nuclear Regulatory Inspectorate of Ukraine, in its last daily update on Tuesday, reported that there were no violations of safe operation limits at any plants.
It said “security divisions and physical protection services are on high alert”.
The Inspectorate provides daily updates at 6am Irish time but up to mid-morning Wednesday, it had not issued a new one.
While air strikes are a concern, Mr Pasyiuk said most incidents at nuclear power plants arose from problems with operating systems.
“You don’t necessarily have to target the reactors themselves because the cut-off of electricity supply causes difficulties for plants to operate.
“They would have some back-up of course but it’s designed only for days. The accidents we saw in other places often related to the systems. So we have a very real danger of a nuclear accident at one of the plants.”
Mr Pasyiuk spoke at an online press briefing by a coalition of Ukrainian environmental organisations that are appealing to similar NGOs throughout Europe for help.
They said the Russians had shelled wind turbines and a solar farm and had a presence at hydroelectricity facilities.
There are conflicting reports about the Zaporizhzhia nuclear power plant, Europe’s biggest. Some Russian troops were advancing on it while others said they had already taken it over.
“What is clear is that the Russians are targeting the energy infrastructure as a whole,” said energy expert Kostiantyn Krynytsky.
Work to connect the Ukraine electricity grid with networks in Europe to ensure continued power has begun, and EU energy ministers agreed on Monday to release 60 million barrels of oil from joint national reserves to Ukraine to help keep energy systems functioning.
The NGOs said the ultimate aim must be to end fossil fuel use.
“Global fossil fuel addiction feeds Putin’s war machine,” said Svitlana Romanko, a Ukranian climate activist.
“Fossil fuels are a weapon of mass destruction and we need a non-proliferation treaty ending their use.”
Eighteen students of Trinity College Dublin who are studying on an exchange programme in Russia are receiving “one-to-one” support from the college as the war in Ukraine intensifies.
This is amid a ‘Do Not Travel’ advisory from the Irish Government to the Russian Federation, given the recent Russian invasion of Ukraine.
A college spokesperson told Independent.ie that Trinity is providing students with the supports they need, “including assistance with any travel arrangements”.
The college said it is working with the students “on a one-to-one basis” and asked how students would return to Ireland from Russia, Trinity said it could not “comment on individual students’ situations”.
Tánaiste Leo Varadkar on Tuesday said there are roughly 50 Irish students currently studying in Russia.
The Department of Foreign Affairs (DFA) has said that there are “severely limited or no flight routes to the EU”, and Irish citizens wishing to return to Ireland via air “are advised to book flights via Istanbul, Dubai, or Abu Dhabi”.
The DFA is advising against all travel to Russia at the present moment, and advised all Irish citizens temporarily in Ukraine to leave as “quickly as possible”, given the geopolitical tension between Russia and the West.
“In light of rapidly deteriorating flight corridors, Irish citizens in Russia on a temporary basis may soon find themselves without viable routes back to Ireland,” the DFA said.
The no travel advisory is due to the “rapidly deteriorating flight and transport options in and out of the country”.
The Department also advised Irish citizens in Russia to be “vigilant about their own safety” and “avoid mass gatherings” due to civil unrest in many cities.
The United States is "very open" to imposing sanctions on Russia's oil and gas industry, White House spokeswoman Jen Psaki said on Wednesday, adding that it is weighing the possible impact on global markets and US energy prices.
Asked if Washington and its Western allies would slap sanctions on Moscow's vast energy sector, Psaki told MSNBC in an interview: "We’re very open."
"We’re considering it. It’s very much on the table, but we need to weigh what all of the impacts will be," she added.
Although the United States has not yet targeted Russian oil sales as part of its sweeping economic sanctions on Russia over its invasion of neighboUring Ukraine, U.S. traders have already acted to put such imports on hold, disrupting energy markets.
Also today, European Union diplomats have approved new sanctions against Belarus for its supporting role in Russia's invasion of Ukraine, the French presidency of the EU said on Wednesday.
EU diplomats approved new sanctions against Belarusian people who are playing a role in the attacks to Ukraine, the presidency said on Twitter.
Sanctions will also hit "some economic sectors, and in particular timber, steel and potash," it said in a statement.
An EU official said this week that one aim of the new sanctions on Minsk was to stop exports of any further Belarusian goods to the EU, on top of those already subject to EU sanctions imposed after President Alexander Lukashenko crushed protests following elections in August 2020.
"These measures will be published in the Official Journal of the EU for entry into force," the presidency said in its statement, without indicating the exact timing of the publication.
Some of the sanctions are expected to close loopholes of existing restrictive measures.
The EU is already banning Belarus' exports of potash, a fertilizer made of potassium, and oil products. But diplomats said Belarus was still exporting potash to the EU via Ukraine, and has also boosted its exports to the EU of oil products obtained from coal.
Meanwhile, Ukraine claims its armed forces have foiled an assassination plot against President Volodymyr Zelensky.
Officials said that a unit of elite Chechen special forces, known as Kadyrovites, were involved in the plot against the country’s leader and had been “eliminated.”
Mr Zelensky said after Vladimir Putin’s invasion of Ukraine that he was Russia’s top target for assassination, and warned that “sabotage groups” were already in in Kyiv hunting for him and his family.
“We are well aware of the special operation that was to take place directly by the Kadyrovites to eliminate our president,” said Ukrainian National Security and Defense Council chief Oleksiy Danilo.
He added that the Chechens had been divided into two, with one group eliminated in Gostomel and the other “under fire.”
And he said that Ukrainian authorities were tipped off about the plot by members of Russia’s Federal Security Service, who he claimed do not support the invasion.
Human Rights Watch said it documented a cluster bomb attack outside a hospital in Ukraine’s east in recent days. Local residents also reported the use of such weapons in Kharkiv and the village of Kiyanka, The Kremlin denied using cluster bombs.
If the allegations are confirmed, that would represent a new level of brutality in the war and could lead to even further isolation of Russia.
Unbowed by Western condemnation, Russian officials made new threats of escalation, days after raising the specter of nuclear war. A top Kremlin official warned that the West’s “economic war” against Russia could turn into a “real one”.
More than a half-million people have fled Ukraine, and countless others have taken shelter underground. Bomb damage to water pipes and other basic services have left hundreds of thousands of families without drinking water, UN humanitarian coordinator Martin Griffiths said.
“It is a nightmare, and it seizes you from the inside very strongly. This cannot be explained with words,” said Kharkiv resident Ekaterina Babenko, taking shelter in a basement with neighbours for a fifth straight day.
"“We have small children, elderly people, and frankly speaking it is very frightening.”
A Ukrainian military official said Belarusian troops joined the war yesterday in the Chernihiv region in the north. But just before that, Belarusian President Alexander Lukashenko said his country had no plans to join the fight.
Low-cost European airline Wizz Air has announced it will be offering 100,000 free seats on short-haul flights for Ukrainian refugees.
The Hungary-based airline said on social media that it was "committed" to helping Ukrainian refugees reach their destination.
The seats will be available on Wizz Air flights departing from Poland, Slovakia, Hungary and Romania during the month of March, the airline said.
Meanwhile, Ryanair has announced it will be the first airline to return to Ukraine.
Chief executive Michael O’Leary said it will return to the eastern European country “when it’s safe to do so” but warned that might not be until the winter.
The airline was previously the largest to serve Ukraine.
Ukrainian airspace was closed last week following Russia’s invasion.
Speaking at a press conference in the City of London, Mr O’Leary said: “We do want to see the Ukrainians succeed.”
He added that the best way to punish Russia would be for the West to “drive down the price of oil”.
“We’ve cancelled all Ukrainian routes until the end of March,” he said.
“We do not believe it will be possible to fly to Ukraine for the foreseeable future.”
The “first thing” the Russians did when they invaded was “disable the flight systems” at major airports, he added.
He went on: “We will be the first airline to return to Ukraine when it’s safe to do so.
“But I suspect it will take probably maybe into next winter before those technologies can be restored at Ukrainian airports, when hopefully the Ukrainians will have seen off the Russians and sent them back to where they came from.”
Mr O’Leary said bookings across all routes on Thursday and Friday last week were down by “about 20pc” compared with a week earlier.
Over the weekend and on Monday they were “down about 10pc.”.
He added: “I think it will not have a dramatic impact on bookings as long as the war doesn’t escalate and spread elsewhere.”
The airline has seen a “significant surge” in bookings to and from Poland, as people fleeing Ukraine are “reuniting with family and friends” across Europe.
Mr O’Leary noted that many people fleeting Ukraine have relatives “dotted across Europe”.
US President Joe Biden vowed in his first State of the Union address on Tuesday night to check Moscow’s aggression in Ukraine as he announced a ban on Russian planes in US airspace.
Mr Biden also noted the US was working to seize the yachts and apartments of Russian oligarchs, saying: “We are coming for your ill-begotten gains.”
Mr Biden, in remarks before Congress, highlighted the bravery of Ukrainian defenders and the resolve of a newly reinvigorated Western alliance that has worked to rearm the Ukrainian military and cripple Russia’s economy through sanctions.
He warned of costs to the American economy, as well, but warned ominously that without consequences, Russian President Vladimir Putin’s aggression would not be contained to Ukraine.
“Throughout our history we’ve learned this lesson – when dictators do not pay a price for their aggression, they cause more chaos,” Mr Biden said. “They keep moving. And the costs and threats to America and the world keep rising.”
As Mr Biden spoke, Russian forces were escalating their attacks in Ukraine, having bombarded the central square of country’s second-biggest city Kharkiv and Kyiv’s main TV tower, killing at least five people. The Babi Yar Holocaust memorial was also damaged.
Many politicians wore pins on their lapels honouring Ukraine.
Even before the Russian invasion sent energy costs skyrocketing, prices for American families had been rising, and the Covid-19 pandemic continues to hurt families and the country’s economy.