Burnham becomes the UK's seventh PM in a decade after Starmer resigns
Burnham became Britain's new prime minister on Monday after King Charles III formally asked him to form a government at Buckingham Palace.
The ceremony, known as "kissing hands", marks the formal transfer of power.
Although the monarch officially appoints the prime minister, the role is ceremonial.
The transfer of power came after Burnham replaced Keir Starmer as leader of the governing Labour Party on Friday.
Under Britain's parliamentary system, a governing party can change its leader, and therefore its prime minister, without holding a general election.
Burnham, 56, is Britain's seventh prime minister since 2016.
"This is a moment for reflection and new resolution," Burnham said outside 10 Downing Street after taking office.
New direction
He vowed to put forward a 10-year plan for Britain, saying the country needed to make politics work.
"Britain needs to show the world that we can regain our stability once again," Burnham said.
He pledged to end rough sleeping, improve living standards and make politics "less toxic". He also promised a "circuit breaker" for a political and economic system he said had been heading in the wrong direction since the 1980s.
The former mayor of Greater Manchester was the only candidate in Labour's leadership contest, winning the backing of 379 of the party's 403 MPs.
Attention will now turn to the ministers he appoints to his Cabinet, with changes expected to some of the government's most senior posts.
Burnham likely to steer steady ship on UK foreign policy
Starmer: 'My work is done'
Burnham succeeds Keir Starmer, who resigned after losing the support of his own party following a series of political setbacks.
Before travelling to Buckingham Palace to tender his resignation, Starmer defended his record in office.
"My work is done," he said, adding that Britain was "stronger and fairer than it was two years ago" when he became prime minister.
"I go with good grace, I go with a smile, and I go proud of everything that we have achieved," he added.
Starmer led Labour from its heavy election defeat in 2019 to a landslide victory in the 2024 general election.
During his final appearance in parliament, he pointed to his government's support for Ukraine, stronger employment rights, measures to reduce child poverty and higher defence spending among its achievements.
On 14 July, French President Emmanuel Macron awarded Starmer France's top honour, the Legion d'honneur, at the end of a meeting of the so-called Coalition of the Willing that the two leaders set up to help Ukraine following Russia's invasion.