The Kauffman skein algebra of a surface was originally defined to be a straightforward generalization of the Kauffman bracket polynomial for knots. Only later was it realized as a quantization of the PSL_2 C character variety of the surface. The Kauffman skein algebra thus emerged as an important connector between quantum topology and hyperbolic geometry. In this talk, we’ll describe how to construct representations of the Kauffman skein algebra and how to construct invariants to help tell them apart. This is joint work with F. Bonahon.