Let G be a locally compact group, and let L1 (G) be the Banach algebra which is the group algebra of G. We consider a variety of Banach left L1 (G)-modules over L1 (G), and seek to determine conditions on G that determine when these modules are either projective or injective or flat in the category. The answers typically involve G being compact or discrete or amenable. For example, in the case where G is discrete and 1 < p < ∞, we find that the module ℓp (G) is injective whenever G is amenable, and that, if it is amenable, then G is ‘pseudo-amenable’, a property very close to that of amenability.