These lectures review some aspects of the theory of moduli spaces which have recently become important in string theory. They begin with some elements of complex geometry. There follows a general description of moduli space, followed by its complex structure. This lets us ask about the analytic properties of the string integrand and ultimately gives its intrinsic form. Along the way we discuss the holomorphic factorization theorems. Finally we discuss infinities in string theory and their interpretation in terms of the boundary of moduli space, following Belavin and Knizhnik.