The size-Ramsey number of a graph G is the smallest number of edges in a graph 56256;56320; with the Ramsey property for G, that is, with the property that any colouring of the edges of 56256;56320; with two colours (say) contains a monochromatic copy of G. The study of size-Ramsey numbers was proposed by Erd733;os, Faudree, Rousseau, and Schelp in 1978, when they investigated the size-Ramsey number of certain classes of graphs and, among others, raised some questions concerning the size-Ramsey number of paths. In this talk, we shall survey some results that have been discovered since, focusing on a couple of recent results obtained by the study of Ramsey properties of fairly sparse random graphs by means of the regularity lemma.