The topology of spaces of continuous maps has been a topic of great interest for decades. Meanwhile, through the Weil conjectures, the topology (specifically, l-adic cohomology) of certain moduli spaces of algebraic functions has had a profound impact on number theory. In this talk, I will outline a framework for studying the cohomology of the moduli space of algebraic maps Alg(X, Y) under certain conditions on X and Y. Our methods, in particular, confirm the geometric Manin's conjecture, which predicts an asymptotic estimate for the number of algebraic maps over finite fields, in the special case where Y is a complete simplicial toric variety and X is a curve. No prior knowledge of algebraic geometry beyond that of a typical first-year graduate student will be assumed.