Conservative leader David Cameron has become the UK's new prime minister after the resignation of Gordon Brown.
Lib Dem leader Nick Clegg will be his deputy after they agreed to the UK's first coalition government in 70 years.
Mr Cameron, who at 43 is the youngest PM in nearly 200 years, vowed to set aside party differences and govern "in the national interest".
Mr Clegg said he acknowledged some Lib Dem voters would have doubts about the deal but urged them to "keep faith".
Mr Cameron's party won the most seats in the general election last week, but not enough to secure an overall Commons majority, resulting in a hung Parliament.
Following hours of talks with the Conservatives on Tuesday, the Lib Dem parliamentary party and its federal executive endorsed the coalition agreement by the required three-quarters majority at a meeting that broke up just after midnight.
Speaking minutes later, Mr Clegg said: "I hope this is the start of the new politics I have always believed in - diverse, plural, where politicians of different persuasions come together, overcome their differences in order to deliver good government for the sake of the whole country."
He acknowledged there would be problems and "glitches" and, in a message to Lib Dem voters, he added: "I can imagine this evening you'll be having many questions, maybe many doubts, about this new governing arrangement.
"But I want to assure you that I wouldn't have entered into this agreement unless I was genuinely convinced that it offers a unique opportunity to deliver the kind of changes you and I believe in.
"So I hope you'll keep faith with us, I hope you will let us prove to you that we can serve you and this country with humility, with fairness at the heart of everything we do, and with total dedication to the interests and livelihoods of everyone in Great Britain."
Meanwhile, details are emerging from Conservative sources about the new government's programme, including:
Mr Cameron has begun the work of appointing his first cabinet, with the Tories' George Osborne as chancellor, William Hague as foreign secretary and Liam Fox as defence secretary.
Mr Clegg's chief of staff, Danny Alexander, who was part of the party's negotiating team, is to be Scottish Secretary, the BBC understands.
Lib Dem Treasury spokesman Vince Cable has been given responsibility for "business and banks" but it is not known if his title will be chief secretary to the Treasury, a senior Lib Dem source said.
A Downing Street spokesman said it had been agreed that five cabinet posts would be filled by Liberal Democrats, including the appointment of Mr Clegg, although there are expected to be about 20 Lib Dems in government jobs in total.
Mr Cameron's arrival in Downing Street marks the end of 13 years of Labour rule.
The coalition is also the first Liberal Democrat and Conservative power-sharing deal at Westminster in history.
Mr Cameron, who is six months younger than Tony Blair was when he entered Downing Street in 1997, is the youngest prime minister since 1812 and the first Old Etonian to hold the office since the early 1960s.
Barack Obama was the first foreign leader to congratulate Mr Cameron in a brief telephone call during which the US president invited the new prime minister to visit Washington in the summer, Downing Street said.
German Chancellor Angela Merkel also offered her congratulations and invited Mr Cameron to visit Berlin.
In a speech outside his new Downing Street home, after travelling to Buckingham Palace to formally accept the Queen's request to form the next government, Mr Cameron paid tribute to outgoing Prime Minister Gordon Brown for his long years of public service.
He also pledged to tackle Britain's "pressing problems" - the deficit, social problems and to "rebuild trust in our political system".
He said he aimed to "help build a more responsible society here in Britain... those who can should and those who can't, we will always help. I want to make sure that my government always looks after the elderly, the frail, the poorest in our country.
"We must take everyone through with us on some of the difficult decisions we have ahead.
"I came into politics because I love this country, I think its best days still lie ahead and I believe deeply in public service.
"I think the service our country needs right now is to face up to our big challenges, to confront our problems, take difficult decisions, lead people through those decisions, so that together we can reach better times ahead."
The Conservatives had been in days of negotiations with the Lib Dems - who also negotiated with Labour - after last Thursday's UK election.
Earlier the Lib Dems said talks with Labour had failed because "the Labour Party never took seriously the prospects of forming a progressive, reforming government".
A spokesman said key members of the Labour team "gave every impression of wanting the process to fail" and the party had made "no attempt at all" to agree a common approach on issues like schools funding and tax reform.
"Certain key Labour cabinet ministers were determined to undermine any agreement by holding out on policy issues and suggesting that Labour would not deliver on proportional representation and might not marshal the votes to secure even the most modest form of electoral reform," he said.
Labour's Lord Mandelson told the BBC they had been "up for" a deal, but the Lib Dems had "created so many barriers and obstacles that perhaps they thought their interests lay on the Tory side, on the Conservative side, rather than the progressive side".
After it became clear the talks had failed, Mr Brown tendered his resignation and said he wished the next prime minister well.
In an emotional resignation statement in Downing Street, Mr Brown thanked his staff, his wife Sarah and their children, who joined the couple as they left for Buckingham Palace.
Mr Brown said it had been "a privilege to serve" adding: "I loved the job not for its prestige, its titles and its ceremony - which I do not love at all. No, I loved the job for its potential to make this country I love fairer, more tolerant, more green, more democratic, more prosperous and more just - truly a greater Britain."
He also paid tribute to the courage of the armed forces, adding: "I will never forget all those who have died in honour and whose families today live in grief."
Later he thanked Labour activists and MPs for all their efforts and told them Labour's general election performance was "my fault, and my fault alone".
Mr Brown had announced on Monday that he would step down as Labour leader by September.