In the first part of this talk I will discuss the construction of parameter and moduli spaces associated to families of rational functions, including the use of mulitplier systems, an explicit description of the moduli space of rational maps of degree two, and a sampling of open questions concerning such spaces. The second part will focus on the dynamical modular curves that classify quadratic polynomials having a specified periodic point structure, analogous to classical elliptic modular curves. The third part will describe relations between the field of definition and the field of moduli of a rational map.