  We define a Banach algebra A to be dual if $A = (A_\ast)^\ast$ for a closed
submodule $A_\ast$ of $A^\ast$. The class of dual Banach algebras includes all
$W^\ast$-algebras, but also all algebras M(G) for locally compact groups G, all
algebras L(E) for reflexive Banach spaces E, as well as all biduals of Arens
regular Banach algebras. The general impression is that amenable, dual Banach
algebras are rather the exception than the rule. We confirm this impression. We
first show that under certain conditions an amenable dual Banach algebra is
already super-amenable and thus finite-dimensional. We then develop two notions
of amenability - Connes-amenability and strong Connes-amenability - which take
the $w^\ast$-topology on dual Banach algebras into account. We relate the
amenability of an Arens regular Banach algebra A to the (strong)
Connes-amenability of $A^{\ast\ast}$; as an application, we show that there are
reflexive Banach spaces with the approximation property such that L(E) is not
Connes-amenable. We characterize the amenability of inner amenable locally
compact groups in terms of their algebras of pseudo-measures. Finally, we give
a proof of the known fact that the amenable von Neumann algebras are the
subhomogeneous ones which avoids the equivalence of amenability and nuclearity
for $C^\ast$-algebras.
