Given a graph H and a set of graphs F , let ex(n, H, F) denote the maximum possible number of copies of H in an F-free graph on n vertices. We investigate the function ex(n, H, F), when H and members of F are cycles. Let C k denote the cycle of length k and let C k = {C 3 , C 4 ,. .. , C k }. We highlight the main results below. (i) We show that ex(n, C 2l , C 2k) = Θ(n l) for any l, k ≥ 2. Moreover, in some cases we determine it asymptotically: We show that ex(n, C 4 , C 2k) = (1 + o(1)) (k−1)(k−2) 4 n 2 and that the maximum possible number of C 6 's in a C 8-free bipartite graph is n 3 + O(n 5/2). (ii) Erdős's Girth Conjecture states that for any positive integer k, there exist a constant c > 0 depending only on k, and a family of graphs {G n } such that |V (G n)|= n, |E(G n)|≥ cn 1+1/k with girth more than 2k. Solymosi and Wong [38] proved that if this conjecture holds, then for any l ≥ 3 we have ex(n, C 2l , C 2l−1) = Θ(n 2l/(l−1)). We prove that their result is sharp in the sense that forbidding any other even cycle decreases the number of C 2l 's significantly: For any k > l, we have ex(n, C 2l , C 2l−1 ∪ {C 2k }) = Θ(n 2). More generally, we show that for any k > l and m ≥ 2 such that 2k = ml, we have ex(n, C ml , C 2l−1 ∪ {C 2k }) = Θ(n m). (iii) We prove ex(n, C 2l+1 , C 2l) = Θ(n 2+1/l), provided a stronger version of Erdős's Girth Conjecture holds (which is known to be true when l = 2, 3, 5). This result is also sharp in the sense that forbidding one more cycle decreases the number of C 2l+1 's significantly: More precisely, we have ex(n, C 2l+1 , C 2l ∪ {C 2k }) = O(n 2− 1 l+1), and ex(n, C 2l+1 , C 2l ∪ {C 2k+1 }) = O(n 2) for l > k ≥ 2. (iv) We also study the maximum number of paths of given length in a C k-free graph, and prove asymptotically sharp bounds in some cases.