  For a family of graphs $\F$, a graph is called $\F$-free if it does not
contain any member of $\F$ as a subgraph. The generalized Tur\'an number
$\ex(n,K_r,\F)$ is the maximum number of $K_r$ in an $n$-vertex $\F$-free graph
and $\ex(n,K_2,\F)=\ex(n,\F)$, i.e., the classical Tur\'an number. Let
$M_{s+1}$ be a matching on $s+1$ edges and $F$ be any graph. In this paper, we
determine $\ex(n,K_r, \{M_{s+1},F\})$ apart from a constant additive term and
also give a condition when the error constant term can be determined. In
particular, we give the exact value of $\ex(n,\{M_{s+1},F\})$ for $F$ being any
non-bipartite graph or some bipartite graphs. Furthermore, we determine
$\ex(n,K_r,\{M_{s+1},F\})$ when $F$ is color critical with $\chi(F)\ge
\max\{r+1,4\}$.
  These extend the results in [2,11,18].
