  Let $G$ be a nontrivial edge-colored connected graph. An edge-cut $R$ of $G$
is called a rainbow cut if no two edges of it are colored the same. An
edge-colored graph $G$ is rainbow disconnected if for every two vertices $u$
and $v$, there exists a $u-v$ rainbow cut. For a connected graph $G$, the
rainbow disconnection number of $G$, denoted by $rd(G)$, is defined as the
smallest number of colors that are needed in order to make $G$ rainbow
disconnected. In this paper, we first solve a conjecture that determines the
maximum size of a connected graph $G$ of order $n$ with $rd(G) = k$ for given
integers $k$ and $n$ with $1\leq k\leq n-1$, where $n$ is odd, posed by
Chartrand et al. in \cite{CDHHZ}. Secondly, we discuss bounds of the rainbow
disconnection numbers for complete multipartite graphs, critical graphs,
minimal graphs with respect to chromatic index and regular graphs, and give the
rainbow disconnection numbers for several special graphs. Finally, we get the
Nordhaus-Gaddum-type theorem for the rainbow disconnection number of graphs. We
prove that if $G$ and $\overline{G}$ are both connected, then $n-2 \leq
rd(G)+rd(\overline{G})\leq 2n-5$ and $n-3\leq rd(G)\cdot rd(\overline{G})\leq
(n-2)(n-3)$. Furthermore, examples are given to show that the upper bounds are
sharp for $n\geq 6$, and the lower bounds are sharp when $G=\overline{G}=P_4$.
