The original Shafarevich conjecture states that for any given (not
necessarily
projective) curve B and for any given genus q, the number of non-isotrivial
smooth
families of curves of genus q is finite and there are no such families
unless
2g(B)-2+d>0 where d is the number of points needed to be added to B to make
it
projective, i.e., d=#(B'\B) where B' is a smooth projective curve containing
B as an
open subset.

In this talk I will discuss recent results towards various higher
dimensional
generalizations of this conjecture. These include results of joint efforts
with
various subsets of {Daniel Greb, Stefan Kebekus, Max Lieblich}.