Solutions of the Yang-Baxter equation

We give the basic definitions connected with the Yang-Baxter equation (factorization condition for a multiparticle $S$-matrix) and formulate the problem of classifying its solutions. We list the known methods of solution of the Y–B equation, and also various applications of this equation to the theory of completely integrable quantum and classical systems. A generalization of the Y–B equation to the case of $Z_2$-graduation is obtained, a possible connection with the theory of representations is noted. The supplement contains about 20 explicit solutions.