In a statement, Kensington Palace said:
The Duke and Duchess of Cambridge are delighted to announce that they have named their son George Alexander Louis.
The baby will be known as His Royal Highness Prince George of Cambridge.
The Duke and Duchess of Cambridge have named their son George Alexander Louis, Kensington Palace said.
The Duke and Duchess of Cambridge will announce the name of their newborn baby shortly, ITV News understands.
The Duke and Duchess of Cambridge and their son have arrived at the Middleton family home in Bucklebury.
Timmy Mallett returned to Windsor today to add the latest edition to the royal family to a portrait of the Duchess of Cambridge.
Timmy decorated a phonebox with paintings of the Queen, the Duchess of Cambridge and Prince Harry last year.
Today he added a crown, a pram and a blue ribbon on the Duchess of Cambridge's portrait, displayed on the side of the 'Ring-a-Royal-Phonebox'.
Around half an hour after the visit from the Queen, William, Kate and their baby were seen being driven away from Kensington Palace.
The Duchess was sitting in the back of the vehicle next to the baby seat while the Duke was a front-seat passenger.
Prince Harry has visited the Duke and Duchess of Cambridge and seen their new baby, Kensington Palace has said.
The Duke and Duchess of Cambridge have left Kensington Palace with their newborn son. It is unclear at this stage where the royal couple and the prince are travelling to.
The Queen made the short journey from Buckingham Palace to visit the Duke and Duchess of Cambridge after the birth of their son yesterday. The visit comes before she heads to Balmoral in Scotland for her traditional summer break on Friday.
Dressed in a turquoise floral outfit, the Queen looked relaxed as she left Kensington Palace after spending just over half an hour with the Duke and Duchess of Cambridge's first child.
She had travelled to see William and Kate and their baby without the Duke of Edinburgh who is still convalescing at Windsor Castle following exploratory abdominal surgery last month.