It is well-known that certain matrix solutions of the braid or Yang-Baxter equations lead to braided categories, knot invariants, quantum groups and other important constructions. However, these equations are also very interesting at the level of set maps $r:X\times X\to X\times X,$ where X is a set and r is a bijection, a line of study proposed by Drinfeld. Solutions extend linearly to very special linear solutions but also lead to a great deal of combinatorics, and to algebras with very nice algebraic and homological properties including those relating to Artin-Schelter regularity, Koszulity, being a PBW Noetherian domain, and the existence of noncommutative Groebner bases. In this talk I will examine solutions mainly from the point of view of finite permutation groups: a solution gives rise to a map from X to the symmetric group $Sym(X)$ on X satisfying certain conditions. Our results include many new constructions based on strong twisted union and wreath product, with an investigation of retracts and the multipermutation level and the solvable length of the groups defined by the solutions; and new results about decompositions and factorisations of the groups defined by invariant subsets of the solution. This talk is based on a joint work with Peter Cameron.