We shall discuss a theorem of Bernstein published in 1975 about the number of common solutions of n complex polynomials in n variables in terms of the mixed volumes of their Newton polytopes. This is a far reaching generalisation of the Fundamental Theorem of Algebra and Bezoutâ€™s Theorem about intersections of plane algebraic curves. If time permits, we shall sketch a proof of Bernsteinâ€™s theorem using Hilbert functions of monomial ideals in polynomial rings.