We study the problem of counting cycles in the adjacency list streaming model, fully resolving in which settings there exist sublinear space algorithms. Our main upper bound is a two-pass algorithm for estimating triangles that uses wtO(m/T^2/3) space, where m is the edge count and T is the triangle count of the graph. On the other hand, we show that no sublinear space multipass algorithm exists for counting ell-cycles for ell geq 5. Finally, we show that counting 4-cycles is intermediate: sublinear space algorithms exist in multipass but not single-pass settings.