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
Erd&#733;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.