Given an arbitrary (bridgeless) network G=(V, E), we develop a distributed algorithm which runs on the network itself to find a small cycle cover for the network, using only local knowledge and message passing. A cycle cover C of G is a set of cycles such that every edge of E is contained in at least one cycle in C. Although the minimum cycle cover problem is conjectured to be N P-complete, our algorithm guarantees a small cycle cover of O(m + n log n) size, where n and m are the number of processors V and communication links E in the network respectively. In an asynchronous bounded delay network where message length is O(log n) bits, the algorithm requires O(n log n) time and O(mn + n 2 log n) messages in the worst case.