New Prime Minister David Cameron has said his "historic" Conservative-led coalition government will be united and provide "strong and stable" leadership.
In a good-humoured press conference with Lib Dem leader Nick Clegg, who is now deputy PM, he said they would "take Britain in a historic new direction".
Their agenda was to cut the deficit, support troops, clean up politics and build a "stronger society".
Mr Clegg acknowledged "big risks" but pledged a "bold, reforming government".
He is among five Liberal Democrats appointed to Cabinet posts, something Mr Cameron said showed "the strength and depth of the coalition and our sincere determination to work together constructively".
The coalition is the first time the Conservatives and Liberal Democrats have had a power-sharing deal at Westminster and the first coalition in the UK since the Second World War.
Mr Cameron's arrival in Downing Street marks the end of 13 years of Labour rule. The Conservative leader, who is six months younger than Tony Blair was when he won power in 1997, is the youngest prime minister since 1812 and the first Old Etonian to hold the office since the early 1960s.
The Conservatives won the most seats in last week's general election, but not enough to secure an overall Commons majority, resulting in a hung Parliament.
After days of talks with the Lib Dems - the UK's third biggest party - a deal was reached on Tuesday that resulted in Labour Prime Minister Gordon Brown resigning.
In a joint press conference in the Downing Street garden, Mr Cameron said the coalition government could mark a "historic and seismic shift" in British politics.
He said they had discussed having a minority Conservative government, supported by the Lib Dems on key votes but had concluded that was "uninspiring".
Instead the two parties had decided to go for a full coalition to be "an administration united behind three key principles - freedom, fairness and responsibility.
"And it will be an administration united behind one key purpose and that is to give our country the strong and stable and determined leadership that we need for the long-term."
He said the government would "take Britain in a historic new direction, a direction of hope and unity, conviction and common purpose".
Rising unemployment figures were another sign "of the economic mistakes of the past decade" and said no government in modern times had been left with "such a terrible economic inheritance".
Mr Clegg admitted both party leaders were taking "big risks" but said it would be a "new politics": "It's a new kind of government, a radical, reforming government where it needs to be and a source of reassurance and stability at a time of great uncertainty in our country too."
Both laughed off differences between the parties and animosity in the past - Mr Cameron apologising after a past description of Mr Clegg as a joke was brought up by a reporter.
He said they wanted to make it work adding: "If it means swallowing some humble pie, and it means eating some of your words, I can't think of a more excellent diet."
Mr Cameron has now completed appointing his first cabinet, with Lib Dem Vince Cable becoming Business Secretary, George Osborne Chancellor, William Hague Foreign Secretary and Theresa May Home Secretary.
Other appointments include Ken Clarke as Lord Chancellor and Justice Secretary, Liam Fox as Defence Secretary and Andrew Lansley as Health Secretary.
Mr Clegg's chief of staff, Danny Alexander, who was part of the party's negotiating team, is to be Scottish Secretary. Lib Dem David Laws will be Chief Secretary to the Treasury.
There are expected to be about 20 Lib Dems - more than a third of their MPs - in government jobs in total.
Meanwhile, details of the policy agreement between the parties have been published, including:
The Lib Dem parliamentary party and its federal executive endorsed the coalition agreement by the required three-quarters majority shortly after midnight.
Mr Clegg acknowledged some Lib Dem voters would have doubts but told them: "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."
Mr Hague told BBC Radio 4's Today programme a five-year fixed-term parliament had been introduced to help guarantee stability.
The only way to remove a government before the five years was up would be in a vote of confidence backed by 55% of MPs. Current rules are that the votes of 50% of MPs, plus one, can remove a government. The Conservatives currently have 47% of MPs.
US President Barack Obama was the first foreign leader to congratulate Mr Cameron in a brief telephone call. Others include German Chancellor Angela Merkel, Indian PM Manmohan Singh, Australian PM Kevin Rudd, French President Nicolas Sarkozy and Chinese premier Wen Jiabao.
After it became clear Labour could not reach an agreement with the Lib Dems that would have allowed them to remain in power, Mr Brown tendered his resignation, saying it had been a privilege to serve "this country I love".
He stepped down as Labour leader with immediate effect - deputy leader Harriet Harman will take over until a leadership contest is held.
Former home secretary Alan Johnson has already ruled himself out in favour of former foreign secretary David Miliband, who has announced he will be a leadership candidate.