The paper surveys the current state of the theory of permutation representations of _nite quasigroups. A permutation representation of a quasigroup includes a Markov chain for each element of the quasigroup, and yields an iterated function system in the sense of fractal geometry. If the quasigroup is associative, the concept specializes to the usual notion of a permutation representation of a group, the transition matrices of the Markov chains becoming permutation matrices in this case. The class of all permutation representations of a given _xed quasigroup forms a covariety of coalgebras. Burnside's Lemma extends to quasigroup permutation representations. The theory leads to a new approach to the study of Lagrangean properties of loops.