The talk will start by recalling essential dimension and looking at some elementary examples. For certain kinds of Deligne-Mumford stacks, P. Brosnan Z. Reichstein and A. Vistoli have proved a very powerful genericity theorem which reduces the calculation of essential dimension to that of gerbes. For Gm-gerbes there is a conjectural formula of J.L Colliot-Thelene, N. Karpenko and A. Merkurjev for the essential dimension of a gerbe that relates the essential dimension to Brauer invariants such as the index. In joint work with I. Biswas and N. Hoffmann, modulo this conjecture, the essential dimension of the moduli stack of vector bundles on a smooth projective curve was computed. Recently these results were extended to orbifold curves with D. Valluri.