  We investigate the higher-dimensional amenability of tensor products $\A \ptp
\B$ of Banach algebras $\A$ and $\B$. We prove that the weak bidimension $db_w$
of the tensor product $\A \ptp \B$ of Banach algebras $\A$ and $\B$ with
bounded approximate identities satisfies \[ db_w \A \ptp \B = db_w \A + db_w
\B. \] We show that it cannot be extended to arbitrary Banach algebras. For
example, for a biflat Banach algebra $\A$ which has a left or right, but not
two-sided, bounded approximate identity, we have $db_w \A \ptp \A \le 1$ and
$db_w \A + db_w \A =2.$ We describe explicitly the continuous Hochschild
cohomology $\H^n(\A \ptp \B, (X \ptp Y)^*)$ and the cyclic cohomology
$\H\C^n(\A \ptp \B)$ of certain tensor products $\A \ptp \B$ of Banach algebras
$\A$ and $\B$ with bounded approximate identities; here $(X \ptp Y)^*$ is the
dual bimodule of the tensor product of essential Banach bimodules $X$ and $Y$
over $\A$ and $\B$ respectively.
