Thurston proved that a fibered 3--manifold is hyperbolic if and only if the monodromy is pseudo-Anosov. I'll describe this amazingly rich class of homeomorphisms of surfaces and explain how the fibered 3-manifolds they define provide powerful tools for analyzing their dynamical properties. Then I'll discuss a parallel theory which is developing for ``train track maps'' representing automorphisms of free groups, and the free-by-cyclic groups they define. Different parts of this talk represent joint work with different people, including Dowdall, Farb, Kapovich, and Margalit.