Abstarct: In this thesis we study the geometry of Lie groups with bi-invariant Randers metric. We first give a necessary and suffcient condition for a left invariant Randers metric to be of Berwald type. We then prove that biinvariant Randers metrics are of Berwald type. Finally a necessary and suffcient condition that left invariant Randers metrics on Lie groups are bi-invariant is given.