This model is out of the class of all previously known models and raises a host of unsolved questions which are related to some of the most intractable problems of algebraic geometry which have been with us for 150 years. The chiral Potts models are used to understand the commensurate-incommensurate phase transitions. For N = 3 and 4, the integrable case was discovered in 1986 in Stony Brook and published the following year.