The institute, which focuses on economics, was set up as a non-metropolitan counterweight to London-based think-tanks.
Mr Perman said: "This is a very important and exciting time to be taking over. There is a real hunger for objective research and impartial analysis, which is obviously crucial in the run up to the (independence) referendum, but will not stop with the vote." Mr Perman is chairman of the James Hutton Institute, which conducts scientific research, and a former board member of Scottish Enterprise.
Mr Peat, who leaves after nine years in the post, is a columnist for The Herald. He is chairman of the Royal Zoological Society of Scotland and sits on Scottish Enterprise's board.
He is to become a adviser to the University of Strathclyde's International Public Policy Institute.
Chairman Professor Hector MacQueen said: "Jeremy will be a hard act to follow. He has cemented the reputation of the David Hume Institute for taking an objective, non-partisan, evidence-based, informed and sceptical approach to consideration of the important issues facing Scotland."
The institute was founded 29 years ago by Professor Sir Alan Peacock to complement the work of the pro-free market Institute of Economic Affairs.