Video report by ITV News Political Editor Robert Peston
The Democratic Unionist Party has signed a deal at Downing Street to support a minority Conservative government after several weeks of post-election negotiations.
The prime minister hailed the power pact deal as a "very, very good one" and insisted her party shares "many values" with the Northern Irish MPs.
DUP leader Arlene Foster said she was "delighted" at an arrangement that she said secures £1.5bn in new spending power in return for support in the House of Commons.
Opposition figures have condemned the cash injection as a "bung" and a "shoddy little deal".
Under a "supply and confidence" arrangement, the DUP guarantees its 10 MPs will vote with the government on the Queen's Speech, the Budget, and legislation relating to Brexit and national security.
Why are May and Foster's names not on the deal?
The leaders stopped for a photograph after shaking hands before entering Number 10 to end talks on Monday morning that had begun in the days after Mrs May's government was returned to power without a majority.
However, the official paperwork was signed by Tory chief whip Gavin Williamson and his DUP equivalent Sir Jeffrey Donaldson.
Mrs May and Mrs Foster, along with First Minister Damian Green and DUP deputy leader Nigel Dodds, watched on.
Mr Green later confirmed to ITV News political editorRobert Peston the whips' signature meant the DUP pact would continue even if Mrs May stood down as PM because it is an agreement between the parties and not the leaders.
What have May and Foster said about the deal?
Mrs May, who had met her new allies on the steps outside Downing Street, stressed their newfound unity as the deal was reached at Number 10.
"We share many values in terms of wanting to see prosperity across the UK, the value of the union, the important bond between the different parts of the United Kingdom," the PM said at Number 10.
"We very much want to see that protected and enhanced and we also share the desire to ensure a strong government, able to put through its programme and provide for issues like the Brexit negotiations, but also national security issues.
"So the agreement we have come to is a very, very good one, and (we) look forward to working with you."
Mrs Foster said: "We're delighted that we have reached this agreement, which I think works, obviously, for national stability."
What are the key parts of the deal?
The agreement, which covers the length of the Parliament but will be reviewed after each session, includes:
An extra £1 billion in funding will go to Northern Ireland (added to new flexibilities on existing spending powers)
A pledge not to change the triple-lock on pensions (ending a Tory manifesto pledge to downgrade to a double-lock) and an axeing of the Tory-proposed means-testing for the winter fuel allowance
The Nato commitment to spend at least 2% of national income on defence will be met
The Armed Forces Covenant will be introduced "throughout the United Kingdom". The DUP previously claimed Northern Ireland had been left out
Agreement that the Conservative party will "never be neutral in expressing its support" for the Union but pledges to "govern in the interests of all parts of the community" in Northern Ireland
Acceptance from the DUP it will have "no involvement" in the Government's political talks in Northern Ireland and recognises the "need for early restoration of inclusive and stable" devolved government.
Read the deal in full:
What's included in the £1.5bn funding?
Mrs Foster confirmed the promise of support from the DUP's 10 MPs secured her country "financial support of £1bn in the next two years and new flexibilities on almost £500m previously committed to Northern Ireland".
The extra £1 billion funding, which covers two years, breaks down as follows:
£400m for infrastructure projects
£200m for improvement of health service
£150m for ultra-fast broadband
£100m to address immediate pressures in health and education
£100m for tackling deprivation (over five years)
£50m for mental health services (over five years)
What is the reaction of opposition figures?
The arrangement, in particular the funding commitments, was swiftly criticised by opposition parties and the first ministers of Wales and Scotland.
Scottish Labour leader Kezia Dugdale echoed the sentiment, saying it was "vital that all nations and regions of the UK also get extra funding to end austerity" and challenging Scottish Tory leader Ruth Davidson to secure it.
Welsh First Minister Carwyn Jones said the £1bn was a "straight bung" that undermined the government's recent promise for fair funding for the nations and regions, while Plaid Cymru said Wales should receive around £1.7bn.
In Northern Ireland, SDLP leader Colum Eastwood accused the Tories of having "bought the DUP" and warned "any position which attempts to wrestle power back from a local executive will be opposed in the strongest possible terms".
In Westminster, Labour leader Jeremy Corbyn said the government must explain the source of its funding and confirm if all parts of the UK will receive the equivalent.
"This Tory-DUP deal is clearly not in the national interest but in May's party's interest to help her cling to power," he added.
Lib Dem leader Tim Farron accused Mrs May of throwing "cash at 10 MPs in a grubby attempt to keep her Cabinet squatting in Number 10".
Green Party co-leader Jonathan Bartley meanwhile attacked Mrs May's claim of "shared values" between the Conservatives and DUP as "alarming", adding: "Which is she referring to? Their opposition to equal marriage, abortion or climate action?"
Similar criticism of the deal had come from Tory grandee Lord Chris Patten, who told ITV's Peston on Sunday the alliance would be politically costly for the Conservatives because the DUP are "toxic".
First Secretary of State Damian Green later made a statement to the House of Commons on the deal, paving the way for angry questions from the opposition benches.
Labour's Emily Thornberry, called the deal "shabby" and "reckless", warning it risked wrecking the fragile Good Friday Agreement.
"For the government to be putting such an agreement in jeopardy just to prop up this dismal prime minister is nothing short of a disgrace."
Much of the concern, though, was over the deal's £1.5bn price tag. Pete Wishart of the SNP called it a "pathetic, grubby little deal demonstrating all the worst excesses of pork barrel politics".
Many, like Paula Sherriff, Labour MP for Dewsbury, wanted to know how the government could set that kind of sum aside and still justify cuts.
"Hospitals schools and other public services in my constituency continue to face unprecedented cuts," she noted. "Is there any money in the magic money tree for the Dewsbury constituency?"
Plaid Cymru MP Liz Saville Roberts called the deal a "bribe" to the DUP, asking where the money for Wales was "in this so-called family of equals".
But sounding a note in defence of the deal that his party had struck with the Tories, DUP deputy leader Nigel Dodds told the House all the "outrage" MPs were voicing over the deal was simply "hypocrisy of the highest order".
Why was the deal done now?
The PM faced a countdown to settle an arrangement to give her a working majority, days before her minority government attempts to get its Queen's Speech plans approved by the Commons.
Mrs Foster was quoted as saying a deal was "imminent" before she arrived for the talks in central London.
The warm welcome was in clear contrast to the last time the DUP leader and her deputy Nigel Dodds drove in for Downing Street talks, when neither the prime minister nor an official came to meet them upon arrival.
How will the deal affect Northern Ireland power talks?
Mrs Foster has suggested the agreement with the Conservatives could trigger a breakthrough in separate negotiations with Sinn Fein over Northern Ireland's collapsed power-sharing executive.
Despite Sinn Fein's public opposition to the Tory-DUP deal, Mrs Foster believes the potential financial benefits to Northern Ireland could bring about goodwill with the investment relying on a working executive at Stormont.
The DUP leader said in Downing Street she was "determined" to see the executive "back in place as soon as possible".