In the past few years there has been much study of random two dimensional surfaces. These provide simple models of string theories with a few degrees of freedom, as well as toy models of quantum gravity. They have possible applications to the statistical mechanics of phase boundaries and to the development of an effective string description of QCD.

Recently, methods have been developed to treat these theories nonperturbatively, based on discrete triangulations of the surfaces that can be generated by simple matrix models. Exact solutions with a rich mathematical structure have emerged. All these matters are discussed fully in this book.Contents:

• Large N Limit and Discretized Two-dimensional Quantum Gravity (E Brézin)
• Quantum Gravity as Dynamical Triangulation (A A Migdal)
• Introduction to the Statistical Mechanics of Random Surfaces and membranes (F David)
• Non-perturbative Effects in two Dimensional Quantum Gravity (F David)
• The c = 1 Matrix Models (D J Gross)
• Singular States in 2D Quantum Gravity (A Polyakov)
• Topological Field Theory and 2D Quantum Gravity (R Dijkgraaf)

Readership: Theoretical, particle and mathematical physicists.