Much of extremal graph theory has concentrated either on finding very small subgraphs of a large graph (such as Turan's theorem [Turan, P., On an extremal problem in graph theory (in Hungarian), Matematiko Fizicki Lapok 48 (1941), 436–452]) or on finding spanning subgraphs (such as Dirac's theorem [Dirac, G.A., Some theorems on abstract graphs, Proc. London Math. Soc. s3-2 (1952), 69–81] or more recently work of Komlos, Sarkozy and Szemeredi [Komlos, J., G. N. Sarkozy and E. Szemeredi, On the square of a Hamiltonian cycle in dense graphs, Random Struct. Algorithms 9 (1996), 193-211; Komlos, J., G. N. Sarkozy and E. Szemeredi, Proof of the Seymour Conjecture for large graphs, Ann. Comb. 2 (1998), 43–60] towards a proof of the Posa-Seymour conjecture). Only a few results give conditions to obtain some intermediate-sized subgraph. We contend that this neglect is unjustified. To support our contention we focus on the illustrative case of minimum degree conditions which guarantee squared-cycles of various lengths, but also offer results, conjectures and comments on other powers of paths and cycles, generalisations thereof, and hypergraph variants.