  The integral over the U(N) unitary group $I=\int DU \exp\Tr A U B U^\dagger$
is reexamined. Various approaches and extensions are first reviewed. The second
half of the paper deals with more recent developments: relation with integrable
Toda lattice hierarchy, diagrammatic expansion and combinatorics, and on what
they teach us on the large $N$ limit of $\log I$.
