The study of Anosov (and pseudo-Anosov) flows on 3-manifolds is a beautiful interface of dynamics, geometry and topology: each flow gives rise to two invariant transverse 2-dimensional foliations, as well as the 1-dimensional foliation of the manifold by orbits, and the study of these foliations can help us classify flows.
Starting with geodesic flow on the unit tangent bundle of a surface, I will describe some of the ways we construct many examples on manifolds with given geometry/topology, and recent work towards the classification problem.  This involves joint work (in various iterations) with Barthelmé, Bowden, and Frankel.