  We present an exact solution of the q-state Potts model on a class of
generalized Sierpinski fractal lattices. The model is shown to possess an
ordered phase at low temperatures and a continuous transition to the high
temperature disordered phase at any q>=1. Multicriticality is observed in the
presence of a symmetry-breaking field. Exact renormalization group analysis
yields the phase diagram of the model and a complete set of critical exponents
at various transitions.
