The greatest common divisor may be defined and exists, more generally, for multivariate polynomials over a field or the ring of integers, and also over a unique factorization domain. There exist algorithms to compute them as soon as one has a GCD algorithm in the ring of coefficients. These algorithms proceed by a recursion on the number of variables to reduce the problem to a variant of the Euclidean algorithm. They are a fundamental tool in computer algebra, because computer algebra systems use them systematically to simplify fractions. Conversely, most of the modern theory of polynomial GCD has been developed to satisfy the need for efficiency of computer algebra systems.