- William jokes: 'He's got her looks, thankfully'
- Kate: 'It's very emotional, it's such a special time. I think any parent will know what this feeling feels like'
- The Duke and the Duchess spent their first night at home with their new son
- Prince Charles describes royal baby boy as 'marvellous' after his visit
- Birth of baby, weighing 8lbs 6oz, was announced by Kensington Palace on Monday night
- Kensington Palace say announcement on baby's names to be made 'in due course'
- 'George' is bookmaker's favourite for boy's name, closely followed by James
Smiling proudly as a happy family, this is the moment the Duke and Duchess of Cambridge introduced their newborn baby boy to the world.
Kate and William, both 31, posed with their royal prince on the steps of the Lindo Wing early on Tuesday evening, with baby Cambridge just over one day old.
The Duchess held her son first and the couple looked relaxed and smiled broadly as the world's media captured the moment.
William walked forward with Kate to answer a few questions. He began by joking: "He's got a good pair of lungs on him, that's for sure. He's a big boy, he's quite heavy. We are still working on a name so we will have that as soon as we can."
He added: "It's the first time we have seen him really so we are having a proper chance to catch up."
The Duke and the Duchess spent their first night at home with their new son in their Kensington Palace apartment.
[All the latest royal baby news]
The Duchess, who still had a bump under her summer dress, said: "It's very emotional, it's such a special time. I think any parent will know what this feeling feels like."
Her husband agreed: "It's very special."
To the assembled media he said: "I'll remind him of his tardiness when he's a bit older. I know how long you've all been standing here so hopefully the hospital and you guys can all go back to normal now and we can go and look after him."
Well-wishers cheered as the couple then drove to Kensington Palace with their baby.
William smiled from behind the wheel and Kate, who was sitting on the back seat, waved with her son next to her in a baby seat.
Earlier in the day, William's father the Prince of Wales said he was "thrilled and very excited" following his visit with the Duchess of Cambridge to see the birth of his first grandchild.
A clearly thrilled Mrs Middleton described her grandson, the future king, as "absolutely beautiful".
Carole and Michael Middleton spent more than an hour with their daughter Kate and son-in-law William, and a smiling Mrs Middleton, speaking about mother and baby, said: "They are both doing really well, and we are so thrilled."
When asked by the world's media, who are camped outside, how the proud parents were doing, she replied: "Fabulously."
Asked what the first cuddle with her grandson was like, Mrs Middleton, who had been ushered forward by her husband to speak to the press, said: "Amazing, It's all coming back."
But when asked if she had suggested a name for the baby, she said: "Absolutely not, but thank you."
It was the first visit the royal couple and their son have received since the newborn was delivered yesterday at 4.24pm, weighing 8lb 6oz.
The Middletons arrived in a black cab and were greeted on the steps of the Lindo Wing by its lead clinician Jonathan Ramsay, before they walked up to the doorway.
The couple turned and smiled broadly as they posed for a few seconds for the media camped opposite the building, but did not reply to the many questions shouted out by journalists.
Attention has now turned to seeing the royal prince for the first time, as the whole country waits to find out his name.
After the birth William and Kate expressed their gratitude to the medical team at the Lindo Wing after their son was born at the unit at 4.24pm on Monday, weighing 8lb 6oz.
The royal couple said in a statement: "We would like to thank the staff at the Lindo Wing and the whole hospital for the tremendous care the three of us have received.
The easel outside Buckingham Palace makes the announcement of the Duchess's newborn baby boy. (PA)
"We know it has been a very busy period for the hospital and we would like to thank everyone - staff, patients and visitors - for their understanding during this time."Hundreds of well-wishers had spontaneously celebrated outside Buckingham Palace on Monday night after the official announcement came through.
After the birth, William telephoned his family to tell them the good news, speaking personally to the Queen, his father Charles and younger brother, Prince Harry.
Prince Charles released a touching statement saying he was 'overjoyed' by the news of the royal baby boy, who will be third in line to the throne.
The Prince of Wales said that he was "enormously proud and happy to be a grandfather for the first time", adding that it was "an incredibly special moment for William and Catherine".
Soon after the palace announcement was made, a notice giving details about the baby - the third-in-line to the throne - left St Mary's Hospital in London by car for Buckingham Palace.
It has been signed by the Queen's former gynaecologist Marcus Setchell, who led the medical team that delivered Kate's baby.
The notice was placed on an easel in the palace's forecourt, just like the announcement of William's birth on June 21, 1982 - a traditional element of theatre in marked contrast to the modern age of emails and Twitter.
He will be the 43rd sovereign since William the Conqueror if, as expected, it follows reigns by the Prince of Wales then William.
The Prince of Wales issued a brief statement: "Both my wife and I are overjoyed at the arrival of my first grandchild. It is an incredibly special moment for William and Catherine and we are so thrilled for them on the birth of their baby boy.
"Grandparenthood is a unique moment in anyone's life, as countless kind people have told me in recent months, so I am enormously proud and happy to be a grandfather for the first time and we are eagerly looking forward to seeing the baby in the near future."
Gun salutes sounded across London on Tuesday to mark the birth of the royal baby as the armed forces join in the celebrations.
The King's Troop Royal Horse Artillery, wearing full dress uniform, rode past Buckingham Palace to Green Park where they staged a 41-gun royal salute.
They rode from their forward mounting base in Wellington Barracks into Green Park, where 71 horses pulled six First World War-era 13-pounder field guns into position for the royal salute at 2pm.
The Honourable Artillery Company (HAC), the City of London's Army Reserve Regiment, also fired a 62-gun salute from Gun Wharf at the Tower of London at 2pm.
Speaking outside 10 Downing Street, David Cameron said: "It is wonderful news from St Mary's, Paddington and I am sure that right across the country, and indeed right across the Commonwealth, people will be celebrating and wishing the royal couple well.
"It is an important moment in the life of our nation but, I suppose, above all it is a wonderful moment for a warm and loving couple who have got a brand new baby boy.
"It has been a remarkable few years for our royal family - a royal wedding that captured people's hearts, that extraordinary and magnificent jubilee and now this royal birth - all from a family that has given this nation so much incredible service."
"They can know that a proud nation is celebrating with a very proud and happy couple tonight."
American president Barack Obama and his wife Michelle, who has developed a close relationship with the Queen, also spoke of their joy at the birth.
The US leader said: "Michelle and I are so pleased to congratulate the Duke and Duchess of Cambridge on the joyous occasion of the birth of their first child.
"We wish them all the happiness and blessings parenthood brings. The child enters the world at a time of promise and opportunity for our two nations.
"Given the special relationship between us, the American people are pleased to join with the people of the United Kingdom as they celebrate the birth of the young prince."