### Abstract:

In this thesis we discuss amenability properties of the Banach algebra-valued continuous functions on a compact Hausdorff space X. Let A be a Banach algebra. The space of A-valued continuous functions on X, denoted by C(X,A), form a new Banach algebra. We show that C(X,A) has a bounded approximate diagonal (i.e. it is amenable) if and only if A has a bounded approximate diagonal. We also show that if A has a compactly central approximate diagonal then C(X,A) has a compact approximate diagonal. We note that, unlike C(X), in general C(X,A) is not a C*-algebra, and is no longer commutative if A is not so. Our method is inspired by a work of M. Abtahi and Y. Zhang. In addition to the above investigation, we directly construct a bounded approximate diagonal for C0(X), the Banach algebra of the closure of compactly supported continuous functions on a locally compact Hausdorff space X.