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  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.                      
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