Let A be a Banach algebra. In this thesis we want to analysis various notions on certain respectively. Set C B = sup{? b? ? , ? a? A ? }. Define C and M similarly. We are interested to know when (B ? B, ? .? B ? B ) can be viewed as a algebra of A ? A. The main problem to overcome is to determine when B ? B can be embeded into A ? A. However, by passing to a quotient space, we can avoid this downfall. Moreover, we define the Segal algebras. Segal algebras are dense ideais of group algebras of locally compact groups, which constitute Banach algebras with respect to some norms and have some homogeneous structures. Since H. Reiter introdued this notion in 1965, many interesting and important results on Segal algebras have been accumulated. It is interesting that some properties of group algebras are hereditary in Segal algebras, but other’s are not. Segal algebras may be regarded as generalizations of group algebras. On the other hand, a generalization of the notion of Segal algebras to a notion on more general Banach algebras are attempted by J. T. Burnham and otheres. It is well known that every Segal algebra on G is an Segal algebra, but we show that the converse is not true. In addition, we proof that every Segal algebra has an approximate identity and having L -norm equal to 1. Furthermore, we investigate the multipliers and the ideal of this that Cigler put forward in his paper the normed ideals in L (G). At last but not least, we will study amenability particulary approximately weakly amenability of Segal algebras. Beside we prove that if G is a compact group, then for the Segal subalgebra L? (G) of L (G). The notions of approximate weak amenability and weak amenability are equivalent to finiteness of G. This contradict remark 3.4 of the article approximate weak amenability, derivations and Arens regularity of Segal algebras which was written by Ghahramani and Lau.