  First of all, we establish compactness of continuous mappings of the
Orlicz--Sobolev classes $W^{1,\varphi}_{\rm loc}$ with the Calderon type
condition on $\varphi$ and, in particular, of the Sobolev classes $W^{1,p}_{\rm
loc}$ for $p>n-1$ in ${\Bbb R}^n\,,$ $n\ge 3\,,$ with one fixed point. Then we
give a series of theorems on convergence of the Orlicz--Sobolev homeomorphisms
and on semicontinuity in the mean of dilatations of the Sobolev homeomorphisms.
These results lead us to closeness of the corresponding classes of
homeomorpisms. Finally, we come on this basis to criteria of compactness of
classes of Sobolev's homeomorphisms with the corresponding conditions on
dilatations and two fixed points.
