He said he aimed to form a "proper and full coalition" with the Lib Dems to provide "strong, stable government".
His party won the most seats in the UK general election last week, but not an overall majority.
In a speech outside his new Downing Street home, Mr Cameron said he and Lib Dem leader Nick Clegg would "put aside party differences and work hard for the common good and the national interest".
He paid tribute to outgoing PM Gordon Brown for his long years of public service and said he would tackle Britain's "pressing problems" - the deficit, social problems and to "rebuild trust in our political system".
Mr Cameron stressed there would be "difficult decisions" but said he wanted to take people through them to reach "better times ahead".more..on bbc