Under certain conditions phase transitions in systems with quenched disorder are expected to exhibit a different behaviour than in the corresponding pure system. Here we discuss a series of Monte Carlo studies of a special type of such disordered systems, namely spin models defined on quenched, random lattices exhibiting geometrical disorder in the connectivity of the lattice sites. In two dimensions we present results for the q-state Potts model on random tri-valent (Φ3) planar graphs, which appear quite naturally in the dynamically triangulated random surface (DTRS) approach to quantum gravity, as well as on Poissonian random lattices of Voronoi/Delaunay type. Both cases, q⩽4 and >4, are discussed which, in the pure model without disorder, give rise to second- and first-order phase transitions, respectively. In three dimensions results for the Ising model on Poissonian random lattices are briefly described. We conclude with a comparison of the two types of connectivity disorder with the more standard case of bond disorder, and a discussion of the distinguishing differences.
