UK's new princess spends first night at Kensington Palace
The name of their second child - sister to Prince George - could be announced later and members of the Royal Family and the duchess's parents may visit.
The princess, who is fourth in line to the throne, was delivered at 08:34 BST on Saturday at St Mary's Hospital in London, and weighed 8lbs 3oz (3.7kg).
Tower Bridge was among several London landmarks lit up in pink in her honour.
Trafalgar Square's fountains and the London Eye were also illuminated on Saturday night to mark the birth.
The Cambridges will spend the next few days at Kensington Palace before they are expected to travel to their country home Anmer Hall on the Queen's Sandringham estate in Norfolk.
Bookmakers have said Alice and Charlotte have emerged as the favourite names for the princess, followed by Olivia, Elizabeth, and Victoria.
Meanwhile, almost all of the UK's national newspapers have given over the entire front pages of their Sunday editions to the birth and a photograph of the new princess.
The princess's first public appearance, with her parents, leaving hospital, will be the exception not the rule.
Prince William and his wife will shield the fourth in line to the throne in the same way they have her brother, Prince George.
Her lack of exposure in her early years, will not limit or diminish the global fascination this baby will attract. It's a fascination which began while she was still in the womb.
Soon we'll know her name. The absence of knowledge hasn't stopped the speculation. Elizabeth, Victoria, Alice and Charlotte have all attracted bets at the bookies.
And will Kate and William honour his mother and choose Diana for one of their daughter's names?
The duchess was admitted to the private Lindo Wing at St Mary's Hospital in Paddington, west London, at 06:00 BST on Saturday.
The birth was announced by Clarence House on Twitter at about 11:00 BST, but also with the traditional bulletin on a gilded easel outside Buckingham Palace - a practice that dates to 1837. The easel will remain outside the palace until early afternoon.
With their new daughter wrapped in a white shawl, the duke and duchess emerged from the hospital to crowds of well-wishers and the world's media, a little less than 10 hours after the birth.
They stood on the steps of the Lindo Wing briefly before heading back inside to put the sleeping princess in a car seat. Prince William, who had been present for the birth, then drove them to Kensington Palace.
The couple did not speak to the media as they had done at the time of Prince George's birth in July 2013.
But when Prince William left the hospital shortly before 16:00 BST, to fetch his son for a visit with his new sister, he told those outside he was "very happy".
In a statement issued after the birth, Kensington Palace said: "Their Royal Highnesses would like to thank all staff at the hospital for the care and treatment they have all received.
"They would also like to thank everyone for their warm wishes."
The duke's father the Prince of Wales, who had said he wanted his second grandchild to be a girl, and Duchess of Cornwall were left "absolutely delighted" by the news, Clarence House has said.
William's uncle Earl Spencer said: "It's wonderful news - we are all thrilled for all four of them."
Messages of congratulations also came from the Archbishop of Canterbury Justin Welby and political leaders including the Prime Minister David Cameron, who called it "wonderful news".
US president Barack Obama and First Lady Michelle wished the duke and duchess "much joy and happiness on the occasion of the arrival of the newest member of their family".
On Monday, the princess's birth will be marked by gun salutes in Hyde Park and the Tower of London.
Soldiers from The King's Troop Royal Horse Artillery will ride out in procession from Wellington Barracks, near Buckingham Palace to sound 41 shots in the park at 14:00 BST. A 62 Gun Salute by the Honourable Artillery Company will take place at the Tower of London at the same time.
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