Given a topological surface (or a compact 2d smooth manifold) we can ask which complex manifolds have this as their underlying topological manifold.  There is a complex (in fact Kahler!) manifold, called Teichmuller space which parametrizes these complex manifolds (up to isotopy).  This talk will introduce this space, and some other simple descriptions of it, such as those given by quadratic differentials, and representations of the fundamental group.  Time permitting some more recent work on coordinate charts on analogues of these spaces will be mentioned.