For graphs $G_1$ and $G_2$, the Ramsey number $R(G_1,G_2)$ is the minimum number of vertices in a graph $G$ such that whenever the edges of $G$ are colored red or blue, there will be either a red copy of $G_1$ or a blue copy of $G_2$. Similarly, the size Ramsey number $\widehat R(G_1,G_2)$ is the minimum number of edges in a graph $G$ such that whenever the edges of $G$ are colored red or blue, there will be either a red copy of $G_1$ or a blue copy of $G_2$. It is immediate from these definitions that $$ \widehat R(G_1,G_2) \le {R(G_1,G_2) \choose 2}. $$ \par The problem of determining $\widehat R(K_n,tK_2)$, where $K_n$ is the complete graph on $n$ vertices and $tK_2$ is a matching with $t$ edges, was studied decades ago by P. Erdős and R.~J. Faudree [in {\it Finite and infinite sets, Vol.\ I, II (Eger, 1981)}, 247--264, Colloq. Math. Soc. János Bolyai, 37, North-Holland, Amsterdam, 1984; MR0818238]. They determined this value when $n \ge 4t-1$ and asked whether $$ \lim_{t \to \infty} {\widehat R(K_n,tK_2) \over t \widehat R(K_n,K_2)} = \min \biggl \{{{n+2t-2 \choose 2} \over t {n \choose 2}} \biggm | t \in {\bf N} \biggr\}. $$ \par In the paper under review, the value of $\widehat R(K_n,tK_2)$ is determined for all positive integers $n$ and $t$, and the question above is answered in the affirmative. Generalizations to hypergraphs and to more than two colors are also considered.