Morgan is the son of refugees, his father Arthur Morgenthau was a German Jew and his mother a Polish Catholic.
He studied at the University of Leeds. In 2003 he wrote the screenplay for The Deal, a TV movie, which dealt with the political agreement between Tony Blair and Gordon Brown. 2006 Morgan's first play Frost / Nixon was premiered in Donmar Warehouse: It's about the television interview of the British television journalist David Frost of former U.S. President Richard Nixon. For his screenplay for The Queen (2006, directed by Stephen Frears ), he received the 2007 Golden Globe Award and nominations for an Academy Award and European Film Award. Also in 2006 he wrote the screenplay for The Last King of Scotland, about the Ugandan dictator Idi Amin, with that him the 2007 BAFTA Award earned. For his film script for Ron Howard's film theater Frost / Nixon he received the 2008 Satellite Award in 2009 as well as nominations for a Golden Globe Award and an Oscar.
Morgan lives with his Austrian wife Lila, a daughter of Karel Schwarzenberg in Vienna.