The solution set of a polynomial system decomposes into a union of irreducible components. The set of polynomials imposes on each component a positive integer known as the multiplicity of the component. This number is of interest not only because of its meaning in applications but also because a number of numerical methods have difficulty in problems where the multiplicity of a component is greater than one. In this talk, I will discuss a numerical algorithm for determining the multiplicity of a component of an algebraic set. This is joint work with Chris Peterson and Andrew Sommese.