Abstract: My talk will involve two concepts which are apparently very different. The character variety of a surface S, consisting of homomorphisms from the fundamental group of S to a Lie group G, arises in many different branches of mathematics. The classical Kauffman bracket is an invariant of knots and links in space, closely related to the Jones polynomial. When G = SL_2(C), Turaev showed that the character variety can be quantised by a generalisation of Kauffman brackets to the surface S. I will discuss the classification problem for Kauffman brackets on S, with results, conjectures and interesting examples.