The 25-year-old mother-of-two was found dead at her home in Wrotham, Kent, yesterday after officers were called "following a report of concern for the welfare of a woman", a force spokesman said.
Shortly after the news of the tragic death broke, her father Bob Geldof released a touching tribute in which he relayed how the family were suffering "beyond pain".
The investigation into her death, which is being treated as "non-suspicious, but an unexplained sudden death", is continuing today, with a post mortem and possible cause of death expected this week.
Naming Ms Geldof as the deceased, a Kent police spokesman said: "Officers were called to the 25-year-old's home in Fairseat Lane, Wrotham at 1.35pm on Monday, 7 April 2014. Peaches was pronounced dead at the scene.
"The death is being treated as a non-suspicious, but an unexplained sudden death.
"Officers are working to establish the circumstances and will be compiling a report of their findings for the coroner. It is expected that a post-mortem examination will be carried out in the next few days."
Ms Geldof was a prolific tweeter and the final message she sent on Sunday was a picture of her as a child with her mother, with the message "Me and my mum". She had two young sons with her second husband, musician Tom Cohen.
Bob Geldof said: "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."
The statement was signed Bob, Jeanne, Fifi, Pixie and Tiger Geldof.
Cohen said: "My beloved wife Peaches was adored by myself and her two sons Astala and Phaedra and I shall bring them up with their mother in their hearts every day. We shall love her for ever."
Ms Geldof was just 11 years old when her mother Paula Yates, died from an accidental heroin overdose in 2000, aged 41.
Yates divorced Bob Geldof in 1996 after forming a relationship with INXS frontman Michael Hutchence.
Hutchence was found dead in a hotel room in Sydney, Australia, in 1997, and Yates went on to lose custody of the three daughters she had with Geldof - Peaches, Pixie and Fifi - the following year.
Bob Geldof later adopted Yates and Hutchence's daughter Heavenly Hiraani Tiger Lily.
Ms Geldof's oldest son Astala is 23 months old, while 11-month-old Phaedra was born on what would have been her mother's 54th birthday.
Ms Geldof was 19 when she married US musician Max Drummey at the Little White Wedding Chapel in Las Vegas in August 2008. The couple announced that they had decided to split amicably in February 2009 before divorcing in 2011.
Ms Geldof married Cohen, lead singer of south-east London band SCUM, in September 2012 at the church in Davington, Kent, where her parents married 26 years earlier. It was also where her mother's funeral was held.
A host of celebrities including Phillip Schofield, Holly Willoughby, Ellie Goulding, Simon Cowell and Lily Allen paid tribute.
Model Daisy Lowe posted a picture of a broken heart on Twitter.
Irish President Michael D Higgins, who begins a state visit to Britain today and was due to meet Geldof during the trip, said his thoughts were with the family.
"I extend my deepest sympathies to Bob Geldof and his family on the sudden and untimely death of his daughter Peaches," he said.
"This is such a difficult cross to bear for any family and all of our thoughts are with Peaches' family and friends at this time. Sabina and I were due to meet Bob Geldof while on the state visit and we are thinking of him at this time of immense loss."
As police officers searched the bins, Kimmy Milham, 30, from Maidstone, laid flowers outside the gates of Ms Geldof's house.
She said she did not know her personally, but had a friend whose children attend the same nursery as her children.
"She was a wonderful person and a wonderful mother," she said.
"It's just tragic that her two sons will not grow up with her.
"Everybody at the nursery loved her and we all admired the way she brought up her kids.
"I always read her blog. She will always be in our hearts."