I discuss how systems with non-abelian anyons can be used to build a topological quantum computer. Operations are performed by braiding the anyons, because the outcome of braiding is a purely topological property, such quantum computers should be robust against local errors. I will give several examples of how such anyons arise in fractional quantum Hall systems and in quantum loop models. Mathematical byproducts of this work are algebraic proofs and extensions of Tutte's identities for the chromatic polynomial (the zero-temperature Potts-model partition function).