We discuss some natural maps from a unitary group ${\rm U}(n)$ to a
smaller group ${\rm U}(n-m)$. (These maps are versions of the
Livšic characteristic function.) We calculate explicitly the
direct images of the Haar measure under some maps. We evaluate some
matrix integrals over classical groups and some symmetric
spaces. (Values of the integrals are products of $\Gamma$-functions.)
These integrals generalize Hua integrals. We construct inverse limits
of unitary groups equipped with analogues of Haar measure and evaluate
some integrals over these inverse limits.