May, 59, campaigned for the leadership of the governing centre-right Conservative Party as a safe pair of hands, after spending six years as home secretary, one of the toughest jobs in British politics.
She arrived at her new Downing Street residence with a promise to tackle "burning injustice".
"As we leave the European Union we will forge a bold new positive role for ourselves in the world," she said, flanked by her banker husband, Philip.
"And we will make Britain a country that works not for a privileged few but for everyone of us."
The political disarray caused by the Brexit vote has also sparked a leadership challenge in the main opposition Labour Party.
Its embattled leader Jeremy Corbyn urged May to "abandon the destructive austerity policies which have damaged our economy and undermined living standards".
