  We investigate when the fundamental group of the smooth part of a K3 surface
or Enriques surface with Du Val singularities, is finite. As a corollary we
give an effective upper bound for the order of the fundamental group of the
smooth part of a certain Fano 3-fold. This result supports Conjecture A below,
while Conjecture A (or alternatively the rational connectedness conjecture in
[KoMiMo] which is still open when the dimension is at least 4) would imply that
every log terminal Fano variety has a finite fundamental group (now a Theorem
of S. Takayama).
