Speaking outside 10 Downing Street, she said she would lead a “One Nation” government that would tackle inequality and preserve Scotland’s place in the UK while negotiating the country’s exit from the European Union.
Just an hour earlier, David Cameron bade an emotional farewell to Downing Street alongside his wife Samantha and their three children.
The outgoing prime minister said it had been the “greatest honour” to lead his country and insisted he had left the country “much stronger”. He shared a hug with his family on the steps of Number 10 before going to Buckingham Palace to hand his resignation to the Queen.
Mrs May, only the second woman to become UK Prime Minister, immediately began assembling her Cabinet last night, appointing Phillip Hammond as Chancellor of the Exchequer and making the surprise announcement that Boris Johnson will be Foreign Secretary.
George Osborne, who had been tipped for the Foreign Office, instead joins Mr Cameron on the backbenches, amid claims he was sacked.
Mr Johnson’s appointment represents a remarkable political comeback a fortnight after his own bid to become prime minister was left in ruins.
David Davis, the veteran eurosceptic, will lead crucial Brexit talks with the European Union, heading up a new ministry dedicated to negotiating the UK’s exit deal.
Amber Rudd will become Home Secretary, while Michael Fallon remains in post as Defence Secretary. Brexiteer Liam Fox, who resigned from government in 2011, will head up a new international trade department.
Mrs May’s Cabinet appointments so far balance both sides of the EU referendum debate, and her speech was a clear response to the shock result that propelled her to the door of 10 Downing Street.
Borrowing the Leave campaign slogan that captured the growing anti-establishment and anti-politics mood, she said her government would “do everything we can to give you more control over your lives”.
“When we take the big calls we will think not of the powerful, but you,” Ms May said, addressing the electorate directly. “When we pass new laws we will listen not to the mighty, but to you.
“When it comes to taxes we will prioritise not the wealthy, but you. When it comes to opportunity we won’t entrench the advantages of the fortunate few, we will do everything we can to help anybody, whatever your background, to go as far as your talents will take you.”
She said she understood that life for families who can “just about manage” but lack the security of a reliable income and quality public services was “much more difficult than many people in Westminster realise”.
Ms May said: “I know you are working around the clock, I know you’re doing your best and I know that sometimes life can be a struggle.
“The government I lead will be driven not by the interests of the privileged few, but by yours. We will do everything we can to give you more control over your lives.”
The new Prime Minister praised her predecessor for leading the economy out of crisis, but said her quest against inequality was a continuation of his mission, saying his “true legacy is not about the economy but social justice”.
She said: “From the introduction of same-sex marriage to taking people on low wages out of income tax altogether, David Cameron has led a One Nation government and it is in that spirit that I also plan to lead.
“Because not everybody knows this, but the full title of my party is the Conservative and Unionist Party and that word ‘Unionist’ is very important to me.
“It means we believe in the Union, the precious, precious bond between England, Scotland, Wales and Northern Ireland – but it means something else that is just as important.
“It means we believe in a union not just between the nations of the United Kingdom but between all of our citizens – every one of us – whoever we are and wherever we’re from.
“That means fighting against the burning injustice that if you’re born poor you will die on average nine years earlier than others.
“If you’re black, you are treated more harshly by the criminal justice system than if you’re white. If you’re a white working-class boy, you’re less likely than anybody else in Britain to go to university.
“If you’re at a state school, you’re less likely to reach the top professions than if you’re educated privately. If you’re a woman, you will earn less than a man.
“If you suffer from mental health problems, there’s not enough help to hand. If you’re young, you will find it harder than ever before to own your own home.”
Mr Osborne tweeted about his departure: “It’s been a privilege to be Chancellor these last 6 years. Others will judge – I hope I’ve left the economy in a better state than I found it.”
He added: “A sad, moving day to see my great friend David Cameron leave Downing St. He can be very proud of the job he’s done and service he’s given.”