She arrived at the palace by ministerial car with her husband Philip May, having set off from the House of Commons after Mr Cameron was officially accepted by the Queen.
Mr Cameron advised the Queen to appointed the former Home Secretary in his place.
A spokesperson for the Palace said: "The Queen received in audience Theresa May and requested her to form a new Administration."
In a brief address outside Downing Street before he resigned, Mr Cameron said: "It has been the greatest honour of my life to serve our country as Prime Minister over these last six years and to serve as leader of my party over 11 years.
"And as we leave for the last time, my only wish is continued success for this great country that I love so very much."
Ms May is to give an address outside Downing Street later this afternoon outlining her initial priorities in office.
The 6 most important issues Theresa May needs to address
Show all 6
She is also expected to unveil the core members of her Cabinet later this evening, with other appointments to be made clear in the coming days.
The former Home Secretary, who backed Remain in the European Union referendum debate, will steer Britain's exit from the bloc.
One of her first major milestones in her time as PM will be triggering the so-called "Article 50" in the European Union treaties, which will formally begin the Brexit negotiation process.
Join our commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies
Join our commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies