European Union leaders say they will remain united and strive to protect the bloc's interest following Britain's decision to leave.
In a statement, the leaders said: "The Union will act as one and preserve its interests.
"Our first priority will be to minimise the uncertainty caused by the decision of the United Kingdom for our citizens, businesses and member states."
They said they would "start by focusing on all key arrangements for an orderly withdrawal".
The leaders will meet in Brussels in one month, on April 29.
But a senior European diplomat said that the European Union will not seek to punish Britain for leaving.
There has been speculation in the British press since last year's referendum on Brexit that the other 27 EU members could try to extract maximum suffering from the UK in order to discourage others from leaving.
The diplomat dismissed such views, noting that Britain will have to grapple with the fallout from its departure from the EU's single market.
"Leaving the common market will hurt a lot all on its own," he said.
Meanwhile, French centrist presidential candidate Emmanuel Macron says his priority after Britain's formal request to leave the EU would be to protect the bloc and European interests.
Speaking after meeting with London Mayor Sadiq Khan, Mr Macron said that he also believes Britain and the EU will need to remain close, notably in defence, even if they have to rebuild their relationship.
He also noted that Mr Khan is mayor to 200,000 French citizens living in London and said he would work to determine how Brexit will affect them.
Mr Macron said in English that in "some of them will decide to come back, I will be very happy to host them again, but, obviously, we will work together with the U.K. and we will work together with London because they live there".
Estonia's prime minister added that "we cannot allow ourselves to be paralysed by Brexit" and "we must decisively move on together".
Juri Ratas said that "we accept this notice with regret in our hearts", adding the priority "is to reduce the insecurity of the people and companies that could be influenced by the United Kingdom's withdrawal".
In a separate statement, the Estonian government said that triggering Article 50 ended "uncertainty on the UK's intentions on leaving the EU and the exit's time-frame."
And t he foreign minister of Luxembourg said a "fair solution" following Brexit is in the interest of both the EU and Britain.
Jean Asselborn said in Serbia's capital that "we cannot punish a country that wants to leave the European Union".
But, he added that "when (British Prime Minister) Theresa May says no deal is better than a bad deal I think that's for both sides, not only for one side".
Mr Asselborn explained that "with Brexit the European Union will lose a little bit of money, that's clear, because the contribution of the UK was substantial".
He also warned that "if you are a member of the European Union, you are member of the European Union until the last moment, and you have to fulfil your engagements."
Sweden's prime minister says that he wants "to see organised and result-oriented negotiations" with Britain, saying good relations with London were "important for Britain, for Europe and for Sweden".
Stefan Lofven said Britain had been "a close and valuable partner in the European Union".
Latvian Foreign Minister Edgar Rinkevics tweeted: "As UK has formally triggered Article 50 we should negotiate in a constructive way to forge a fair deal for both EU and UK."
Sweden's ambassador to Britain tweeted that "Swedes in the UK with concerns and questions related to Brexit are welcome to contact Swedish Embassy".
Chancellor Angela Merkel said Germany will push for the impact of Brexit on Germans and other European Union citizens living in Britain to be as minimal as possible.
She said she wants Britain and the EU to remain "close partners".
Ms Merkel said during a speech in Berlin that the divorce talks must first focus on undoing in an orderly fashion four decades of ties between Britain and the EU.
She said: "Only when these questions are cleared up can we subsequently - but hopefully soon - talk about our future relationship."
Ms Merkel also said the remaining EU member nations will negotiate with Britain "in a fair and constructive manner".
She said: "I hope that the British government will also approach the talks in this spirit."
She added that Theresa May had assured her it would in a phone conversation on Tuesday.
The foreign minister of France said Britain's decision to exit the European Union "has lifted a taboo" against what he called "the irreversibility" of the EU.
However, foreign minister Jean-Marc Ayrault also said the letter from the British Government that triggered EU divorce proceedings "has the advantage of bringing clarity".
Mr Ayrault said in a statement that Britain's decision "shall be respected" but that it would be up to the 27 remaining EU members "to implement their common will to strengthen the European Union".
Mr Ayrault cited a vow EU leaders took over the weekend to stick together despite differences as they marked the 60th anniversary of the founding treaty that prefigured the EU.
France's far-right National Front said the EU should work "in a spirit of healthy cooperation" in negotiating Britain's departure from the bloc, and not make the divorce as painful as possible to dissuade other nations from leaving.
French party leader Marine Le Pen, a leading presidential candidate in the election that starts next month, wants her country to leave the European Union and give up the euro currency.
Her party's delegation at the European Parliament said Britain's letter is the start of a "renaissance of national ambition until now trampled on" by European Union leaders.
Meanwhile, the European Parliament's top official for Brexit negotiations wants a deal on the future rights of the three million EU citizens living in Britain and the one million Britons living elsewhere in the bloc before the end of the year.
Parliament Brexit coordinator Guy Verhofstadt said that having that part of the talks ironed out quickly would help provide certainty for EU citizens.
He says the residency agreement could be sealed off as a done deal before it would need to be rubber stamped as part of the overall withdrawal deal after two years.
Mr Verhofstadt said: "It should be a good thing, if we were capable, (to) have an agreement on this before the end of the year so that we can already give that certainty to the citizens even when formally it will be a chapter that will be in the withdrawal agreement."