abstract: In the years 1940-1970, Alexandrov and the "Leningrad School" has developed a rich theory of singular surfaces. These are topological surfaces, with an intrinsic metric for which we can define a notion of curvature which is a Radon measure. This class of surfaces has nice convergence properties and it is remarkably stable with respect to various geometric constructions (such as glueing etc.). It includes polyhedral surfaces and C2 Riemannian surfaces, both classes being dense in the space of Alexandrov surfaces. Any singular surface one can reasonably think of is an Alexandrov surface, yet many geometric properties of smooth surfaces do extend to Alexandrov surfaces. The aim of this lecture is to give a non-technical introduction to Alexandrov's theory, to give examples and some of the fundamental facts from the theory. We will also discuss a classification theorem of (compact) Alexandrov surfaces.