The Jones polynomial is simple to define, yet a mysterious invariant of links in $\mathbb R^3\!$. It extends to links in thickened surfaces, leading to the notion of the skein algebra of a surface, which can be thought of as a quantization of the Teichmüller space. The algebraic structure of skein algebras remains rather mysterious. We will establish some fundamental properties of these objects using the theory of measured foliations and pseudo-Anosov diffeomorphisms of surfaces.