We study the degree-diameter problem for claw-free graphs and 2-regular hypergraphs. Let be the largest order of a claw-free graph of maximum degree Δ and diameter D. We show that , where , for any D and any even . So for claw-free graphs, the well-known Moore bound can be strengthened considerably. We further show that for with (mod 4). We also give an upper bound on the order of -free graphs of given maximum degree and diameter for . We prove similar results for the hypergraph version of the degree-diameter problem. The hypergraph Moore bound states that the order of a hypergraph of maximum degree Δ, rank k, and diameter D is at most . For 2-regular hypergraph of rank and any diameter D, we improve this bound to , where . Our construction of claw-free graphs of diameter 2 yields a similar result for hypergraphs of diameter 2, degree 2, and any even rank .