Reince Priebus, chairman of the Republican National Committee (RNC), will be his chief of staff.
In this role, he will set the tone for the new White House and act as a conduit to Congress and the government.
Stephen Bannon, from the conservative Breitbart News Network, will serve as Mr Trump’s chief strategist.
Mr Bannon stepped aside temporarily as Breitbart’s executive chairman to act as Mr Trump’s campaign chief.
The Republican candidate defeated Democrat Hillary Clinton in Tuesday’s presidential vote, in a result which shocked many, who had expected Mrs Clinton to win following favourable opinion polls.
Mr Trump is due to take over at the White House on 20 January, when Barack Obama steps down after two terms in office.