In this paper we deal with three generalized notions of amenability,
approximate, approximate weak and approximate n-weak amenability. The first two
were introduced and studied by Ghahramani and Loy in [5]. We introduce the
third one. Then we investigate some properties of Banach algebras in each of
these classes. Also we consider approximate weak amenability and approximate
n-weak amenability of second duals.
We introduce the notions of approximate Connes-amenability and approximate
strong Connes-amenability. Then we characterize these two types of dual Banach
algebras in terms of approximate normal virtual diagonal and approximate
$\sigma WC-$ virtual diagonals. Some concrete cases are also discussed.