  Working with a singular variety X, one is often interested in comparing the
set of etale covers of X with that of its smooth locus X_reg. More precisely,
one may ask: What are the obstructions to extend finite etale covers of X_reg
to all of X? How do the etale fundamental groups of X and of its smooth locus
compare? For projective varieties with Kawamata log terminal (klt)
singularities we answer these questions in part. The main result of this paper
asserts that there are no infinite towers of finite morphisms over a klt base
variety where all morphisms are etale in codimension one, but branched over a
small set. In a certain sense, this result can be seen as saying that the
difference between the etale covers of X and those of X_reg is finite if X is
klt. As an immediate application, we construct a finite covering Y of X, etale
in codimension one, such that the etale fundamental groups of Y and Y_reg
agree.
