The authors define the notion of a nondegenerate symmetric set, which is a set \(X\) with an invertible map \(R:X\times X\to X\times X\) satisfying the quantum Yang-Baxter equation \(R^{12} R^{13} R^{23}= R^{23} R^{13} R^{12}\) as maps from \(X\times X\times X\) to \(X\times X\times X\) (where \(R^{ij}\) means \(R\) acting on the \(i\)th and \(j\)th components), and some nondegeneracy and unitarity conditions. A topological interpretation of this notion is given. A structure group \(G_X\) is associated to a nondegenerate symmetric set \(X\), and this is used to prove that nondegenerate symmetric sets, up to isomorphism, are in a bijective correspondence with quadruples \((G,X,\rho,\pi)\), where \(G\) is a group, \(X\) is a set, \(\rho\) is a left action of \(G\) on \(X\), and \(\pi\) is a bijective 1-cocycle of \(G\) with coefficients in \(\mathbb{Z}^X\). Indecomposable nondegenerate symmetric sets with \(p\) elements are classified for prime \(p\), and this is used to show that \(G_X\) is solvable for finite \(X\). Some quantum algebras are associated to nondegenerate symmetric sets by the FRT construction. Several constructions of nondegenerate symmetric sets are presented and the results of a computer calculation which finds all nondegenerate symmetric sets with at most 8 elements are given. Power series solutions of the Yang-Baxter equation are considered.