After a brief review of the history of the star-triangle equation, we shall illustrate its importance with a few results for the two-dimensional Ising model and its generalization to the chiral Potts model. We shall discuss how the recent solutions in terms of higher-genus Riemann surfaces emerge. We shall finish with some further results for the quantities of interest in these new models. More related work is presented in the talks by Profs. Baxter and McCoy.