Abstarct: In this thesis in the first chapters we give the basic concepts of Riemannian and Finsler ge ometry. Then, we study the geometry of two-step nilpotent Lie groups with left-invariant Finsler metric and give an explicit form for the Chern-Rund connection,curvature tensor, Ricci tensor, flag curvature and geodesics of these spaces. Finally, we study The Randers metrics on two-step nilpotent Lie groups of dimension five.