: In this thesis, we present an expanded account of amenability and n-weak amenability for module extension of Banach algebras based on an article by Zhang (2002). Suppose that A is a Banach algebra and X is a Banach A-bimodule. A (continuous) derivation from A into X is a (continuous) linear mapping D : A X which satisfies D(ab)=a . D(b)+D(a) . b for all a,b ? A. The (continuous) derivation D : A X is said to be inner if D(a)=a . x-x . a for some x ? X. Denote by (A,X) the space of all continuous derivations from A into X and by (A,X) the space of all inner derivations from A into X. Then (A,X) is a suace of (A,X). the quotient space (A,X)= (A,X)/ (A,X) is called the first cohomology group of A with coefficients in X. We first discuss the conditions under wich n-weak amenability implies (n+2)-weak amenability for n 0, and the conditions for the unitization of a Banach algebra to be n-weakly amenable. After an extensive study of module extensions of Banach algebras, we give a necessary and sufficient condition for that a Banach algebra of this kind to be n-weakly amenable, separately for odd numbers n and even numbers n. We also discuse the necessary and sufficient conditions for that a Banach algebra A, to be (?, ?)- amenable. These finally offer us a way to construct a counter-example to the question of whether weak amenability implies 3-weak amenability.