Modular forms and modular curves have played a crucial role in modern number theory, but one almost always eventually restricts to considering only forms for congruence subgroups. In this talk I will try to explain how noncongruence subgroups (of SL(2,Z)) fit into the picture. Specifically, for a finite 2-generated group G, I will begin by defining the moduli space of elliptic curves with "G-structures", which will be a congruence modular curve if G is abelian, and noncongruence if G is sufficiently nonabelian. I will describe how this relates to the unbounded denominators conjecture for noncongruence modular forms, and using the work of Scholl, I will then describe a connection between the Fourier coefficients of noncongruence modular forms and Galois actions on the (nonabelian) fundamental groups of punctured elliptic curves. As time allows I will describe a joint work with Deligne showing that metabelian level structures are congruence, and some partial progress towards understanding Hecke operators on nonabelian level structures.