Abstract: We review slope stability for coherent sheaves on algebraic curves and then discuss Tom Bridgeland's generalization of stability to triangulated categories. For a certain triangulated category associated to a Kleinian singularity, Bridgeland conjectured that a connected component of the space of stability conditions should be the universal cover of a $K(G,1)$ for $G$ a generalized braid group and showed this is the case for $G$ the classical braid group. We generalize this to braid groups of types ADE. This is joint work with Hugh Thomas.