The Constraint Satisfaction Problem (CSP) provides a common framework for many combinatorial problems. The general CSP is known to be NP-complete; however, certain restrictions on a possible form of constraints may affect the complexity, and lead to tractable problem classes. There is, therefore, a fundamental research direction, aiming to separate those subclasses of the CSP which are tractable, and those which remain NP-complete.

Schaefer gave an exhaustive solution of this problem for the CSP on a 2-element domain. In this paper we generalise this result to a classification of the complexity of CSPs on a 3-element domain. The main result states that every subclass of the CSP is either tractable and the criterion separating them is that conjectured earlier. We also exhibit a polynomial time algorithm which, for a given set of allowed constraints, outputs if this set gives rise to a tractable problem class. To obtain the main result and the algorithm we extensively use the algebraic technique for the CSP developed by Jeavons and coauthors.