  Let $G$ be an edge-coloured graph. A rainbow subgraph in $G$ is a subgraph
such that its edges have distinct colours. The minimum colour degree
$\delta^c(G)$ of $G$ is the smallest number of distinct colours on the edges
incident with a vertex of $G$. We show that every edge-coloured graph $G$ on
$n\geq 7k/2+2$ vertices with $\delta^c(G) \geq k$ contains a rainbow matching
of size at least $k$, which improves the previous result for $k \ge 10$.
  Let $\Delta_{\text{mon}}(G)$ be the maximum number of edges of the same
colour incident with a vertex of $G$. We also prove that if $t \ge 11$ and
$\Delta_{\text{mon}}(G) \le t$, then $G$ can be edge-decomposed into at most
$\lfloor tn/2 \rfloor $ rainbow matchings. This result is sharp and improves a
result of LeSaulnier and West.
