The Duke and Duchess of Cambridge have spent their first night at home with their new son.
After one night in hospital, William and Kate returned to their Kensington Palace apartment with the baby prince, who is yet to be named.
Showing the new third-in-line to the throne to the world last night, the couple described becoming parents as "such a special time".
The Duke revealed naming the prince was now a priority as he told the press: "We're still working on a name so we will have that as soon as we can.''
Bookmakers William Hill have backed George as the favourite at 2/1, while James is 4/1 and Alexander is 9/1.
William and Kate, looking relaxed and happy, brought their baby out into the public gaze and joked about their newborn being a "big boy".
On the steps of the Lindo Wing of St Mary's Hospital in London the Cambridges posed for the press and waved and smiled at the cheering well-wishers who had gathered to witness the special event.
The baby prince was the star attraction and appeared oblivious to the flashguns of the world's press photographers going off, but did seem to be partially awake during his first public appearance, as his hand poked over the white shawl he was wrapped in.
The Duchess, who still had a little bump under her bespoke Jenny Packham summer dress, held her son first before handing him to the Duke who was casually dressed in dark trousers and a pale blue shirt with his sleeves rolled up.
The royal couple gave an impromptu and good-humoured press conference telling reporters about how their lives had changed.
Asked how he felt, William said: "Very emotional," and the Duchess agreed: ''It's very emotional and such a special time and I think any parent will probably know what this feeling feels like.''
William and Kate's appearance outside the Lindo Wing harked back to the day the Duke emerged into the world outside the same hospital in the arms of the Prince and Princess of Wales in 1982.
But the atmosphere was very different this time, as the royal couple were casually dressed and appeared to be at ease.
Joking about his son William added: ''Well, he's got a good pair of lungs on him, that's for sure. He's a big boy, he's quite heavy."
The new royal baby - the first Prince of Cambridge to be born for more than 190 years - was delivered yesterday at 4.24pm, weighing 8lb 6oz.
Kate revealed that her husband was a hands-on father.
Asked about changing nappies the Duke said: "We've done that already," and the Duchess added: "He's done his first nappy already".
Joking again William said: "He's got her looks, thankfully," while Kate replied: "No, no, I'm not sure about that."
And poking fun at his own thinning hair the Duke said about his son: "He's got way more than me, thank God."
The Duke looked very relaxed holding his son and after speaking to the press for a few minutes returned inside the Lindo Wing with his wife.
The couple soon emerged with their son in a baby seat and William carefully fitted it onto the back seat of a Range Rover beside his wife before driving his family home.
A crowd of well-wishers was waiting to greet the Cambridges when they arrived at Kensington Palace and they cheered as the Duke drove past and smiled from behind the wheel while his wife waved.
In the hours before they left hospital William and Kate received visits from their son's beaming grandparents - Michael and Carole Middleton and the Prince of Wales and Duchess of Cornwall.
Charles's visit to the hospital was a historic moment as it is believed to be the first time three male heirs to the throne - Charles, his son William and grandson - who are expected to reign have come together in more than 100 years.
Asked how his first grandchild was, Charles replied: "Marvellous, thank you very much, absolutely wonderful."