In this thesis, let A and B be two banach algebras and ? be a nonzero character on B. If we equip the A×B with the component addition, scalar multiplication, -norm and ?-Lau product, then with above actions A×B is a Banach algebra denoted by A B and called ?-Lau product Banach algebra. Here, we characterize some notions of amenability as approximate amenability, essential amenability, n-weak amenability and cyclic amenability between A and B and their ?-Lau product. Moreover, for two closed two-sided ideals and of A and B respectively, we investigate the relation between derivations from A into , and derivations from B into with derivations from the A B into under certain conditions. Also, we study the n-ideal amenability of these Banach algebras. Finally, we study Pseudo-amenability, pseudo-contractibility and character pseudo-amenability of A B and their relations with A and B.