We define a new type of search problem called “mutual search”, where players, located on nodes of a completely connected network, are required to locate each other by posing “Anybody at node i?” queries. We give algorithms showing that the minimal number of queries required by two players in a network of n nodes is considerably less than the n − 1 queries one naively expects to be optimal. We give upper and lower bounds around n/2 for the deterministic worst case, sharp to within 5 percent. We also exhibit a simple randomized algorithm whose cost beats the determinstic lower bound, and a deterministic algorithm for k ≥ 2 players with a cost well below n for all k = o(√n). The graph-theoretic framework we formulate for expressing and analyzing algorithms for this problem may be of independent interest. Joint work with Harry Buhrman, Juan Garay, Jaap-Henk Hoepman, Matt Franklin, and John Tromp.