  Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let
$\delta^c(G)$ denote the minimum color-degree of $G$. A subgraph $F$ of $G$ is
called rainbow if any two edges of $F$ have distinct colors. There have been a
lot results in the existing literature on rainbow triangles in edge-colored
complete graphs. Fujita and Magnant showed that for an edge-colored complete
graph $G$ of order $n$, if $\delta^c(G)\geq \frac{n+1}{2}$, then every vertex
of $G$ is contained in a rainbow triangle. In this paper, we show that if
$\delta^c(G)\geq \frac{n+k}{2}$, then every vertex of $G$ is contained in at
least $k$ rainbow triangles, which can be seen as a generalization of their
result. Li showed that for an edge-colored graph $G$ of order $n$, if
$\delta^c(G)\geq \frac{n+1}{2}$, then $G$ contains a rainbow triangle. We show
that if $G$ is complete and $\delta^c(G)\geq \frac{n}{2}$, then $G$ contains a
rainbow triangle and the bound is sharp. Hu et al. showed that for an
edge-colored graph $G$ of order $n\geq 20$, if $\delta^c(G)\geq \frac{n+2}{2}$,
then $G$ contains two vertex-disjoint rainbow triangles. We show that if $G$ is
complete with order $n\geq 8$ and $\delta^c(G)\geq \frac{n+1}{2}$, then $G$
contains two vertex-disjoint rainbow triangles. Moreover, we improve the result
of Hu et al. from $n\geq 20$ to $n\geq 7$, the best possible.
