Packing cycles with modularity constraints Erdos and Posa proved that a there exists a function \$f\$ such that any graph either has \$k\$ disjoint cycles or there exists a set of \$f(k)\$ vertices that intersects every cycle. The analogous statement is not true for odd cycles - there exist numerous examples of graphs that do not have two disjoint odd cycles, and yet no bounded number of vertices intersects every odd cycle. However, Reed has given a partial characterization of when there does not exist a bounded size set of vertices intersecting every odd cycle. We will discuss recent work on when a graph has many disjoint cycles of non-zero length modulo \$m\$ for arbitrary \$m\$. When \$m\$ is odd, we see that again there exists a function \$f\$ such that any graph either has \$k\$ disjoint cycles of non-zero length modulo \$m\$ or there exists a set of at most \$f(k)\$ vertices intersecting every such cycle of non-zero length. When \$m\$ is even, no such function \$f(k)\$ exists. However, the partial characterization of Reed can be extended to describe when a graph has neither many disjoint cycles of non-zero length modulo \$m\$ nor a small set of vertices intersecting every such cycle.