Summary: ``In this paper, we study some cohomological properties of Banach algebras. For a Banach algebra $A$ and a Banach $A$-bimodule $B$, we investigate the vanishing of the first Hochschild cohomology groups $H^1(A^n, B^m)$ and $H^1_{w^*} (A^n, B^m)$, where $0 \leq m$, $n \leq 3$. For amenable Banach algebra $A$, we show that there are Banach $A$-bimodules $C$, $D$ and elements $\germ a, \germ b \in A^**$ such that $$ Z^1(A,C^*)=\{R_{D''(\germ a)}: D\in Z^1(A,C^*)\}=\{L_{D''(\germ b)}:D\in Z^1(A,D^*)\} $$ where, for every $b\in B$, $L_b(a) = ba$ and $R_b(a) =ab$, for every $a\in A$. Moreover, under a condition, we show that if the second transpose of a continuous derivation from the Banach algebra $A$ into $A^*$ i.e., a continuous linear map from $A^{** }$ into $A^{***}$, is a derivation, then A is Arens regular. Finally, we show that if $A$ is a dual left strongly irregular Banach algebra such that its second dual is amenable, then $A$ is reflexive.''