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In this monograph, some new notions of module amenability such as module contractibility, module character amenability and n-weak module amenability for Banach algebras are introduced and some hereditary properties are given. For an inverse semigroup S with subsemigroup E of idempotents, module character amenability of the semigroup algebra l^1(S) is shown to be equivalent to S being amenable. Also, it is proved that l^1(S) is permanently weakly module amenable. The concept of module Arens regularity for Banach algebras and bilinear maps are introduced and they are characterized. The module topological centers of second dual of a Banach algebra are defined and they are found for l^1(S)**. It is proved that l^1 (S)** is module amenable (as an l^1(E)-module) if and only if a maximal group homomorphic image of S is finite. Finally, it is shown under what conditions l^1(S) is module biflat and module biprojective.
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