Smooth rigidity for Anosov flows and geometric applications

The smooth rigidity question in dynamics asks when an a priori weak form of equivalence automatically leads to a much stronger form of equivalence. It turns out that rigidity is common in the context of Anosov flows ---  a continuous conjugacy is automatically smooth. I will survey progress on this question and give several geometric applications to the rigidity of negatively curved manifolds. Based on joint work with F. Rodriguez Hertz.