The properties of the pure-site clusters of spin models, i.e., the clusters which are obtained by joining nearest-neighbor spins of the same sign, are here investigated. In the Ising model in two dimensions it is known that such clusters undergo a percolation transition exactly at the critical point. We show that this result is valid for a wide class of bidimensional systems undergoing a continuous magnetization transition. We provide numerical evidence for discrete as well as for continuous spin models, including
SU(N) lattice gauge theories. The critical percolation exponents do not coincide with the ones of the thermal transition, but they are the same for models belonging to the same universality class.