: In this thesis, we present an expanded account of left character amenability of Banach algebras based on an article by Kaniuth, Lau and Pym (2008). We introduce and study new notation of amenability for Banach algebras. Let A be an arbitrary Banach algebra and ? a homomorphism from A onto C, then A is called ?-amenable if there exists a bounded linear functional m on A * satisfyingm(?)=1 and m(f.a)= ?(a)m(f) for all a A and f A * . We campare left ? –amenability with left amenability of Lau-algebras. It turns out that this concept is considerably more general than that of left amenability for Lau-algebra. Also, we mainly focus on ?-means of norm 1. For example, we show that the existence of such a mean is pointwise property. We discuss some hereditary properties of left ?-amenability Banach algebras. Finally, we study the concept of left character amenability and we show that amenability of G is equivalent to left character amenability of L 1 (G) and M(G).