The definition of the Kauffman bracket skein algebra of an oriented surface was originally motivated by Witten's topological quantum field theory interpretation of the Jones polynomial for knots.  But the skein algebra is also closely related to the SL(2, C)-character variety of the surface, which contains Teichmuller space as a real subvariety.  We'll discuss recent methods for constructing finite-dimensional representations of the skein algebra, and their role in bridging quantum topology and geometric topology.