  This is a survey paper on moduli spaces that have a natural structure of a
(possibly incomplete) locally symmetric variety. We outline the Baily-Borel
compactification for such varieties and compare it with the compactifications
furnished by techniques in algebraic geometry. These differ in general, but we
show that a reconciliation is possible by means of a generalization of the
Baily-Borel technique for a class of incomplete locally symmetric varieties.
The emphasis is here on moduli spaces of varieties other than that of polarized
abelian varieties.
