The Zamolodchikov tetrahedron equation is a fundamental relation for the integrability of 3D lattice models in statistical mechanics, playing a role similar to the Yang-Baxter equation in 2D lattice models. I will discuss recent developments in which new solutions of the tetrahedron equation were constructed with use of quantum cluster algebras and related to 3D supersymmetric gauge theories, and a well-known solution was argued to arise from a brane system in M-theory.