The thesis deals with the problem of finding a characterization of the weak amenability of The weighted grou algebras on non-abelian groups. Also we will show that the weighted group algebras on a discrete and SIN-group are weakly amenable provided that weight is diagonally bounded. Next for every continuous weight we prove the same result for every show that weighted locally compact group algebras. Finally we show that weakly amenable whenever ( , (R) ) is not