I will briefly review some of the major differences between polynomial and transcendental dynamics: multiply connected Fatou components, rates of escape to infinity, wandering domains, local connectedness, and dimensions of Julia sets. I will then introduce and compare the Speiser and Eremenko-Lyubich classes, two collections of entire functions that share some properties with polynomials, but are still broad enough to contain many exotic examples. In a certain sense, Eremenko-Lyubich functions need only satisfy certain simple topological restrictions, while the Speiser subclass must also satisfy geometric constraints. I will then discussquasiconformal folding, a geometric technique for constructing examples in these classes, and describe some of the results that have been obtained using this technique.