The mother-of-two was the daughter of Bob Geldof and Paula Yates.
"Peaches has died," Bob Geldof said this evening in a statement.
"We are beyond pain. She was the wildest, funniest, cleverest, wittiest and the most bonkers of all of us. Writing 'was' destroys me afresh. What a beautiful child.
"How is this possible that we will not see her again? How is that bearable? We loved her and will cherish her forever. How sad that sentence is.
"Tom and her sons Astala and Phaedra will always belong in our family, fractured so often, but never broken. Bob, Jeanne, Fifi, Pixie and Tiger Geldof."
Her husband Tom wrote: "My beloved wife Peaches was adored by myself and her two sons, I shall bring them up with their mother in their hearts everyday. We shall love her forever."
Born in 1989, Peaches began her career writing for Elle magazine at the age of 15.
The following year, she began contributing to the Guardian and Telegraph newspapers.
She also presented television shows including OMG! with Peaches Geldof on ITV2, and worked as a model.
Peaches, who had regularly posted photos of herself and her young family online, tweeted for the last time twenty-four hours ago. The young mother posted a photo of her as a child with her late mother Paula Yates.
Police say they were called to an address in Kent following a report of concern for the welfare of a woman on Monday afternoon.
The woman was later pronounced dead by South East Coast Ambulance Service, and has since been identified as Ms Geldof.
The young mother posted a photo of her as a child with her late mother Paula Yates.
"At this stage, the death is being treated as unexplained and sudden," said a statement from Kent Police.
The second daughter of Bob Geldof, Peaches is married to Scum rocker Tom Cohen.
The couple have two sons together; Astala (2) and Phaedra (eleven months).
Peaches Geldof gave birth to her second son, on what would have been her mum Paula Yates’ 54th birthday, on April 26, 2013.
Peaches elected to give birth via caesarian section last year and is said to have planned it in memory of her mum, who died from a heroin overdose in 2000 aged 41.
She reportedly wanted to give birth the year before on the same date to honour her mum, but Astala arrived some days later.
Peaches was previously married to musician Max Drummey but the pair separated after a six-month marriage in 2009.
The young socialite had struggled with drugs in the past, but police are adamant that a cause of death has not yet been confirmed.
Police have also said they are not treating the death of Ms Geldof as suspicious.