A symbolic algorithm is presented which decomposes a general algebraic set in n dimensions into its irreducible components. Starting with a collection of multivariate defining polynomials, the algorithms produces a set of Gröbner Bases each of which is a collection of defining polynomials for a particular irreducible component. It is proposed that this may be a useful tool in CAGD systems since the degrees of the resulting irreducible components may be smaller than the degree of the original algebraic set. Moreover, the equations for the irreducible components which are produced may indicate special geometric properties that were not apparent from the given equations for the algebraic set. © 1989.