Sophie and the Earl of Wessex visited Windsor Castle as the military paid tribute to Prince Philip with royal gun salutes across the UK and on warships.
They took place in cities including London, Edinburgh and Cardiff, and at Hillsborough Castle in County Down.
All three of the Queen's sons have visited her at Windsor following Prince Philip's death on Friday.
The Duke of York visited the castle on Saturday, and the Prince of Wales travelled there on Friday afternoon.
Military personnel fired 41 rounds at locations across the UK, starting at midday and leaving a one-minute break between each round.
Hundreds of people lined Tower Bridge in London, where members of the Honourable Artillery Company fired rounds from guns facing the Thames.
And the noise of gunfire echoed out from Edinburgh Castle, where more than 100 people gathered as the salute began.
Royal Navy ships at sea, including HMS Diamond and HMS Montrose, also fired salutes in honour of the duke, who served as a naval officer during World War Two and held the office of Lord High Admiral.
Lieutenant Colonel Erica Bridge, who serves with the Royal Artillery, said a royal salute typically involves 21 rounds, but an extra 20 are fired when they are "from designated saluting stations" - such as in this instance.
Speaking to the BBC before the salutes, she said military personnel "will feel great sadness" after the duke's death, and the salutes would "mean a huge amount".
Prince Philip was Queen Elizabeth II's husband of 73 years, and died at Windsor Castle at the age of 99.
Announcing the duke's death on Friday, Buckingham Palace said: "It is with deep sorrow that Her Majesty the Queen announces the death of her beloved husband.
"The Royal Family join with people around the world in mourning his loss."
On Saturday, the Royal Family's Twitter account shared a photograph of the Queen and her husband, along with a quote from a speech she made on their golden wedding anniversary in 1997 in which she described him as having been her "strength and stay all these years".
Reflecting on Prince Philip's life for a BBC programme broadcast on Friday evening, Prince Charles described his father's life as an "astonishing achievement".
First Sea Lord and Chief of the Naval Staff Admiral Tony Radakin, the most senior officer in the Royal Navy, praised the duke's "empathy, affection and engagement" with the fleet.
General Sir Nick Carter, Chief of the Defence Staff, said the duke had been a "great friend, inspiration and role model" for the armed forces.
Similar salutes were fired to mark the death of Queen Victoria in 1901 and Winston Churchill in 1965.
The New Zealand Army will pay tribute in the same way at Point Jerningham in Wellington on Sunday.
Meanwhile, jockeys, owners and trainers marked a two-minute silence in tribute to the duke in the parade ring at Aintree Racecourse, Liverpool.
Flags at the track are at half-mast and jockeys are wearing black armbands.
Final details of the duke's funeral are also expected to be released this weekend.
The funeral will take place at St George's Chapel, Windsor, but the arrangements have been amended in light of the coronavirus pandemic, the College of Arms said in a statement.
The duke will not have a state funeral and there will be no lying-in-state, in line with his wishes, it added.
Members of the public are "regretfully" requested not to attend due to the pandemic, and it is understood the Queen is considering modified funeral and ceremonial arrangements.
The Royal Family has asked people to consider making a donation to a charity instead of leaving flowers in memory of the duke, and an online book of condolence has been launched on the official royal website.
A spokesman for Windsor Great Park said tributes would be removed "respectfully" throughout Saturday and displayed within the castle grounds.
All UK government buildings have been told to fly official flags at half-mast in tribute to the duke until 08:00 on the day after the duke's funeral.
By Becky Morton, BBC News
Windsor evokes memories of royal weddings, with crowds lining the streets waving the union jack. Today the mood is sombre, and some locals - and others from further afield - have come to pay their respects.
Anne Yearsley, 82, from Waltham St Lawrence in Berkshire, has brought her grandchildren Cordelia, 10, and Sherlock, six, to Windsor Castle.
"We just thought it would be wonderful for them to have this memory," she says.
Anne has followed the royal couple's lives since they married in 1947 and remembers their wedding. More recently, she recalls passing the duke riding his carriage in Windsor Great Park a few years ago.
Sherlock is wearing a military uniform for the occasion - passed down from his father who used to wear it as a child watching Trooping the Colour.
The children have been learning about Prince Philip's achievements from news coverage and Cordelia is now excited to do the Duke of Edinburgh Award when she's older.
"I didn't really know much about him before, apart from that he was the husband of the Queen," she says.
"I didn't know he did all these amazing things."
In BBC interviews to mark his death, Prince Philip's children shared their reflections on his life - with Prince Charles calling his legacy an "astonishing achievement".
He said: "His energy was astonishing in supporting my mama [the Queen] - and doing it for such a long time, and, in some extraordinary way, being able to go on doing it for so long.
"What he has done has amounted to an astonishing achievement, I think."
The Princess Royal said the duke "treated everyone as an individual, and gave them the respect he felt they were due as individuals".
A message on the website of the Duke and Duchess of Sussex's non-profit organisation Archewell paid tribute to the "loving memory" of the Duke of Edinburgh, saying: "Thank you for your service... you will be greatly missed."
Meanwhile, Kensington Palace said the Duke of Cambridge would not appear at the Bafta Awards ceremony this weekend.
As Bafta's president - a role that Prince Philip himself formerly held - Prince William had been due to to deliver a speech via video link.
Politicians across the UK also paid tribute to Prince Philip, with Prime Minister Boris Johnson saying the duke had "earned the affection of generations" in the UK and around the world.
Political parties suspended their campaigning for elections on 6 May and parliament will honour the duke on Monday, with the House of Commons sitting at 14:30 for tributes.
International leaders also remembered the duke and sent their condolences to the Queen.
US President Joe Biden and First Lady Jill Biden said in a statement the duke "gladly dedicated himself to the people of the UK, the Commonwealth, and to his family".
The Vatican said Pope Francis was "saddened to learn" of the duke's death, praising his "devotion to his marriage and family, his distinguished record of public service and his commitment to the education and advancement of future generations".
The announcement also saw members of the public gather outside Buckingham Palace and Windsor Castle to pay their respects.
Their first son, Prince Charles, was born in 1948, followed by his sister, Princess Anne, in 1950, Prince Andrew, in 1960, and Prince Edward, in 1964.
Prince Philip was born on the Greek island of Corfu on 10 June 1921.
His father was Prince Andrew of Greece and Denmark, a younger son of King George I of the Hellenes.
His mother, Princess Alice, was a daughter of Prince Louis of Battenberg and a great-granddaughter of Queen Victoria.