  We construct certain subgroups of hyperbolic triangle groups which we call
"congruence" subgroups. These groups include the classical congruence subgroups
of SL_2(ZZ), Hecke triangle groups, and 19 families of arithmetic triangle
groups associated to Shimura curves. We determine the field of moduli of the
curves associated to these groups and thereby realize the groups PSL_2(q) and
PGL_2(q) regularly as Galois groups.
