For a wide variety of Banach algebras A (containing the group algebras L¹(G), M (G) and A(G)) the Arens regularity of A** is equivalent to that A, and the amenability of A** is equivalent to the amenability and regularity of A. In this paper, among other things, we show that this variety contains the weighted group algebras L¹(G, w) and M(G, w).