Triangle groups, the symmetry groups of tessellations of the hyperbolic plane by triangles, have been studied since early work of Hecke and of Klein--the most famous triangle group being $\textrm{SL}_2(\mathbb Z).$ We present a construction of congruence subgroups of triangle groups (joint with Pete L. Clark) that gives rise to curves analogous to the modular curves, and provide some applications to arithmetic. We conclude with some computations that highlight the interesting features of these curves.