Tetrahedron equation and spin integrable models on the cubic lattice

The paper is a review of results on three-dimensional generalization of Yang–Baxter equation obtained starting from the pioneering works of Zamolodchikov (1979) to our works made in spring 1995. The integrability condition for spin statistical models on the simple cubic lattice (tetrahedron equation) is discussed. Different versions of this equation are considered with their symmetrical properties. The solution of the tetrahedron equation corresponding to Bazhanov–Baxter model is considered in detail. The review contains an update list of solutions for this equation. Generalization for unhomogenious spin models with two types of Bolzmann's weights forming the checkerboard lattice is considered.