The senior Tory said last week’s EU referendum had left the country needing "strong leadership and a clear sense of direction".
Despite supporting Britain remaining in the EU, Mrs May ruled out a second referendum on membership of the bloc.
She said: "Brexit means Brexit.
"The campaign was fought, the vote was held, turnout was high and the country gave their verdict.
"There must be no attempts to remain inside the EU, no attempts to rejoin it through the backdoor and no second referendum."
Mrs May promised to put controls on open-door EU migration as part of the Government’s Brexit negotiations.
Even before last week’s historic vote, the Home Secretary had suggested there is a need to end Brussels’ free movement rules.
The Maidenhead MP also ruled out holding a snap general election and an emergency Brexit
Mrs May will face-off against leading Leave campaigners Boris Johnson and Michael Gove in the race for Downing Street.
Work and Pensions Secretary Stephen Crabb, energy minister Andrea Leadsom and ex-defence secretary Liam Fox are also running to be named the new Conservative leader.
The winning candidate will automatically be named Mr Cameron’s replacement in Number 10.
Mrs May was boosted later after Health Secretary Jeremy Hunt announced he will not run for the Tory leadership, despite having previously said he was "seriously considering" a bid.
Throwing his support behing the Home Secretary, Mr Hunt said: "I have decided that now is not the right time for me to run for the leadership - though I remain completely committed to ensuring we secure out position as a great trading nation with sensible controls on migration.
"I believe Theresa May has the strength, judgement and values to deliver those things.
"She is the right choice to lead Britain in a challenging period and will make a truly outstanding Prime Minister."
Education Secretary Nicky Morgan also announced today she will not stand, but put her support behind Mr Gove.
But Scottish Secretary David Mundell gave his backing to Mrs May. He said: "I think in these very difficult and uncertain times we need a serious and competent politician as our leader."
He also took a swipe at Mr Johnson insisting the former London mayor did not have "experience on the world stage", adding: "Playing ping pong at the Olympics is quite different from negotiating Britain's exit from the EU."
Speaking in central London, Mrs May littered her leadership announcement with jibes at Mr Johnson.
In a clear swipe at the ex-London mayor, Mrs May said politics was not a "game".
She said: "If you are from an ordinary working class family, life is just much harder than many people in politics realise.
"You have a job but you don't always have job security, you have your own home but you worry about mortgage rates going up, you can just about manage but you worry about the cost of living and the quality of the local school because there is no other choice for you.
"Frankly, not everybody in Westminster understands what it's like to live like this and some need to be told that it isn't a game.
“It's a serious business that has real consequences for people's lives."
Mrs May made frequent references to her six-year spell at the heart of Government as head of the Home Office, highlighting how she had tackled police corruption, modern slavery and successfully negotiated the deportation of radical cleric Abu Qatada.
She said: “If ever there was a time for a prime minister who is ready and able to do the job from day one, this is it.”
Asked why she thought she was best placed to negotiate a new relationship with the EU outside of the bloc, Mrs May pointed out she had years of experience of negotiating with Brussels leaders.
She said: “I have done this. I have sat round the table. I’ve not just done it I have delivered on negotiations.”
In another dig at Mr Johnson, she recalled his purchase of riot control water cannons before seeking her permission to use them, which she eventually denied.
Mrs May joked: “Boris negotiated in Europe. I seem to remember last time he did a deal with the Germans, he came back with three nearly-new water cannon."
Despite supporting Britain remaining in the EU, Mrs May ruled out a second referendum on membership of the bloc.
She said: "Brexit means Brexit.
"The campaign was fought, the vote was held, turnout was high and the country gave their verdict.
"There must be no attempts to remain inside the EU, no attempts to rejoin it through the backdoor and no second referendum."
Mrs May promised to put controls on open-door EU migration as part of the Government’s Brexit negotiations.
Even before last week’s historic vote, the Home Secretary had suggested there is a need to end Brussels’ free movement rules.
The Maidenhead MP also ruled out holding a snap general election and an emergency Brexit
Mrs May will face-off against leading Leave campaigners Boris Johnson and Michael Gove in the race for Downing Street.
Work and Pensions Secretary Stephen Crabb, energy minister Andrea Leadsom and ex-defence secretary Liam Fox are also running to be named the new Conservative leader.
The winning candidate will automatically be named Mr Cameron’s replacement in Number 10.
Mrs May was boosted later after Health Secretary Jeremy Hunt announced he will not run for the Tory leadership, despite having previously said he was "seriously considering" a bid.
Throwing his support behing the Home Secretary, Mr Hunt said: "I have decided that now is not the right time for me to run for the leadership - though I remain completely committed to ensuring we secure out position as a great trading nation with sensible controls on migration.
"I believe Theresa May has the strength, judgement and values to deliver those things.
"She is the right choice to lead Britain in a challenging period and will make a truly outstanding Prime Minister."
Education Secretary Nicky Morgan also announced today she will not stand, but put her support behind Mr Gove.
But Scottish Secretary David Mundell gave his backing to Mrs May. He said: "I think in these very difficult and uncertain times we need a serious and competent politician as our leader."
He also took a swipe at Mr Johnson insisting the former London mayor did not have "experience on the world stage", adding: "Playing ping pong at the Olympics is quite different from negotiating Britain's exit from the EU."
Speaking in central London, Mrs May littered her leadership announcement with jibes at Mr Johnson.
In a clear swipe at the ex-London mayor, Mrs May said politics was not a "game".
She said: "If you are from an ordinary working class family, life is just much harder than many people in politics realise.
"You have a job but you don't always have job security, you have your own home but you worry about mortgage rates going up, you can just about manage but you worry about the cost of living and the quality of the local school because there is no other choice for you.
"Frankly, not everybody in Westminster understands what it's like to live like this and some need to be told that it isn't a game.
“It's a serious business that has real consequences for people's lives."
Mrs May made frequent references to her six-year spell at the heart of Government as head of the Home Office, highlighting how she had tackled police corruption, modern slavery and successfully negotiated the deportation of radical cleric Abu Qatada.
She said: “If ever there was a time for a prime minister who is ready and able to do the job from day one, this is it.”
Asked why she thought she was best placed to negotiate a new relationship with the EU outside of the bloc, Mrs May pointed out she had years of experience of negotiating with Brussels leaders.
She said: “I have done this. I have sat round the table. I’ve not just done it I have delivered on negotiations.”
In another dig at Mr Johnson, she recalled his purchase of riot control water cannons before seeking her permission to use them, which she eventually denied.
Mrs May joked: “Boris negotiated in Europe. I seem to remember last time he did a deal with the Germans, he came back with three nearly-new water cannon."
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