In Chapter 3 we discussed the properties of the Yang-Baxter equation and the associated quantum R matrix, which are essential ingredients for setting up the algebraic Bethe ansatz. The latter is undoubtedly one of the most significant contributions of the St. Petersburg school to the theory of quantum inverse scattering method. Subsequently a number of variants of the algebraic Bethe ansatz have been developed to deal with situations where the standard technique fails, namely the functional Bethe ansatz, analytical Bethe ansatz, etc.