Video report by ITV News Royal Correspondent Chris Ship
Prince Harry and Meghan have called their newborn son Archie after they proudly introduced him to the world in a photo call at Windsor Castle.
The Duke and Duchess have called their son, Archie Harrison Mountbatten-Windsor, a statement on the couple's Instagram account said.
Harry and Meghan posted a black and white picture on the @SussexRoyal account, showing off Archie to the Queen, Prince Philip and Meghan's mum Doria Ragland.
The post said: "This afternoon Their Royal Highnesses introduced Her Majesty The Queen to her eighth great-grandchild at Windsor Castle. The Duke of Edinburgh and The Duchess' mother were also present for this special occasion."
The Queen and the Duke of Edinburgh were the first members of the Royal family to meet the baby boy at Windsor Castle.
It emerged last night that having revealed Archie to the world, Prince Harry will be away from his new family on Thursday.
He is heading to the Hague in the Netherlands for the one-year countdown to the Invictus Games 2020.
ITV News Royal Editor Chris Ship who was reporting live from Windsor said: ''He [Harry] is still going to the Netherlands for this event to mark next year's Invictus Games. It was something he really didn't want to miss. Of course Meghan does have her mother, Doria, at home with her.''
He then joked: ''And if she's really stuck, we've just checked, the Queen's in, she can always ask her for some advice.''
By deciding to call their son Archie Harrison Mountbatten-Windsor, Harry and Meghan have chosen not to use a title for their first born.
As the first born son of a duke, Archie could have become Earl of Dumbarton - one of Harry's subsidiary titles - or have been Lord Archie Mountbatten-Windsor, but instead he will simply be Master Archie Mountbatten-Windsor.
Archie means "genuine", "bold" and "brave", and is German in origin, short for Archibald, it is now given as a name in its own right.
Harrison means "son of Harry" so is likely to be a tribute to the duke.
The couple looked adoringly at their newborn son as they posed for pictures for the first time as a family of three earlier in the day.
Meghan declared: "It's magic, its's pretty amazing, and I have the best two guys in the world, so I'm really happy."
Speaking in St George's Hall at Windsor Castle as her husband held their two-day-old child, the duchess added: "He has the sweetest temperament, he is really calm and he's been the dream, it's been a special couple of days."
Asked how he found parenting, a clearly delighted Prince Harry said: "It's great. Parenting is amazing.
"It's only been two and a half days, three days, but we're just so thrilled to have our own little bundle of joy and to be able to spend some precious times with him as he slowly, slowly starts to grow up."
When asked who the baby takes after, Meghan said: "We're still trying to figure that out."
Harry added: "Everyone says that babies change so much over two weeks we're basically monitoring how the changing process happens over this next month really. But his looks are changing every single day, so who knows."
The duke joked "he's already got a little bit of facial hair."
An Instagram post was shared on the couple's new sussexroyal account, with three photographs of the duke and duchess and their new child.
It said: "Today The Duke and Duchess of Sussex are delighted to share their first public moment as a family.
"They are so incredibly grateful for the warm wishes and support they’ve received from everyone around the world, since welcoming their son two days ago."
After much anticipation, the royal baby was born at 5.26am on Monday and weighed 7lb 3oz.
Speaking hours after the birth, a beaming Prince Harry said both mother and baby were doing "incredibly well" as he spoke from Windsor Castle's Royal Mews.
The duke revealed the gender of his son, who will be seventh in line to the throne, and said he is "absolutely to-die-for".
He said: "I haven't been at many births, this is definitely my first birth but it was amazing, absolutely incredible.
"I'm so incredibly proud of my wife and as every father and parent would ever say, your baby is absolutely amazing, but this little thing is absolutely to-die-for, so I'm just over the moon."
The couple have chosen to celebrate in private at their new home, Frogmore Cottage, over the past couple of days, before introducing their newborn to the public.
Baby Sussex is the Queen's eight great-grandchild and Prince Charles' fourth grandchild.
The Queen and Prince Philip were "delighted" to hear of the baby's birth on Monday and Doria Ragland, Meghan's mother, was said to be "overjoyed" at the arrival of her first grandchild.
Prince William and Kate congratulated Harry and Meghan on the birth of their first on Tuesday at an event at the Cutty Sark.
William said he was "absolutely thrilled" and welcomed his brother to the "sleep deprivation club that is parenting."
The Queen meanwhile was quizzed about the birth of her new great-grandchild - and her eighth in total.
At a luncheon for members of the Order of Merit, she was asked by a guest "life is good for Your Majesty?"
The smiling Queen replied: "Yes, thank you."
Who was interviewing Meghan and Harry?
Alan Jones has sprung to fame after being chosen as the Royal correspondent to interview Prince Harry on Monday after the birth and again on Wednesday at the photocall in Windsor.
Professionally known as Tony, he said Meghan was "over the moon" at having her husband next to her in "dad mode."
The royal reporter said Harry and Meghan were "very loving and protective" of their bundle of joy.
Mr Jones said: "[Meghan] was quite proud to see her son being held by her husband."
The duchess was "very effusive" about her emotions, adding: "She couldn't keep anything hidden.You could tell she was over the moon having her son and her husband next to her."
He said the stand-out moment of the interview for him was the "touching" way Meghan spoke about her husband and son.
"It's probably her first few words from Meghan, when she said something along the lines of about having 'my guys'.
"Just the phrase 'my guys'... it's very touching.
"It brought it all home how, in a few hours, her life has changed, and now she's a family of three and she's got two people that will mean the world to her.
"And it was just very touching to see that and for her to tell us about it."