There are a number of lattice models, mostly two-dimensional, in statistical mechanics that have been solved exactly by using the star-triangle relation. By this I mean that the partition function per site and the spontaneous magnetization have been calculated in the limit of a large lattice. I shall give a historical overview and discuss the remarkable invariance properties that follow from the star-triangle relation, indicating how they lead to the solutions. The talk is intended for a non-specialist audience.