In this thesis, we present an expanded account of the work done by Dales, Polyakov and Ramsden. In this thesis we shall study general intrinsic properties of projective and injective modules over a Banach algebra. we also prove some hereditary properties. we adopt the convention of studing projective left modules and injective right modules. Let G be a locally compact group, and let L?(G) be the group algebra of G. Dales and Polyakov have recently investigated when varous canonical modules over L?(G)have certain well-known homological properties. we study Banach modules over the measure algebra M(G) for alocally compact group G.