Tetrahedron equation and spin integrable models on a cubic lattice

This paper is a survey of results on the three-dimensional generalization of the Yang-Baxter equation obtained since the pioneer works by Zamolodchikov (1979) up to our articles in 1995. The integrability condition for statistical spin models on a simple cubic lattice (tetrahedron equation) is discussed and different versions of this equation are considered together with their symmetrical properties. The solution of the tetrahedron equation corresponding to the Bazhanov-Baxter model is considered in detail. The review contains an updated list of solutions for this equation. Generalization to inhomogeneous spin models with two types of Boltzmann weights forming a chessboard-type lattice is considered.