Random surfaces and lattice gravity

In this talk I review some of the recent developments in the field of random surfaces and the Dynamical Triangulation approach to simplicial quantum gravity. In two dimensions I focus on the c=1 barrier and the fractal dimension of two-dimensional quantum gravity coupled to matter with emphasis on the comparison of analytic predictions and numerical simulations. Next is a survey of the current understanding in 3 and 4 dimensions. This is followed by a discussion of some problems in the statistical mechanics of membranes. Finally I conclude with a list of problems for the future. (orig.)