We study three-point covers of the projective line whose Galois group is either PSL_2(FF_q) or PGL_2(FF_q). We construct these covers by isolating 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 Shimura curves associated to arithmetic triangle groups. We determine the field of moduli of the curves associated to these groups and thereby realize the above groups regularly as Galois groups in many cases over explicitly given abelian number fields. This is joint work with Pete L. Clark.