Statistical models with a geometrical basis arise in many circumstances, such as the theory of liquids, membranes, polymer networks, defects, microemulsions, interfaces, etc. Gauge theories also lead to random surfaces, as does the theory of extended objects such as strings, and quantum gravity requires a generalization to four-manifolds. From a general point of view, local quantum field theory is rooted in the study of random paths. One may wish to find such a universal model, generalizing Brownian curves, to Brownian manifolds and in the first instance to surfaces (Polyakov, 1981). Despite many efforts, no such universal archetype has been found, although the endeavour towards such a model has uncovered a rich mathematical structure. By necessity, our presentation will be limited to the most elementary aspects.

In the first section we discuss random lattices in Euclidean space. The use of such lattices was advocated as a mean to restore translational invariance while keeping the advantage of a short distance cutoff (Christ, Friedberg and Lee, 1982). The formalism could be generalized to other manifolds, but we refrain from doing to, nor will we pursue the analysis of standard models on random lattices, a difficult subject. Even free field theory on such a lattice opens the Pandora's box of disordered systems. Concepts from random lattices may be useful in the study of liquids or gases.