An edge-colored graph is called rainbow if all the colors on its edges are distinct. Let ${\cal G}$ be a family of graphs. The anti-Ramsey number $AR(n,{\cal G})$ for ${\cal G}$, introduced by Erdős et al., is the maximum number of colors in an edge coloring of $K_n$ that has no rainbow copy of any graph in ${\cal G}$. In this paper, we determine the anti-Ramsey number $AR(n,\Omega_2)$, where $\Omega_2$ denotes the family of graphs that contain two independent cycles.