  The power structure over the Grothendieck (semi)ring of complex
quasi-projective varieties constructed by the authors is used to express the
generating series of classes of Hilbert schemes of zero-dimensional subschemes
on a smooth quasi-projective variety as an exponent of that for the complex
affine space of the same dimension. Specializations of this relation give
formulae for generating series of such invariants of the Hilbert schemes of
points as the Euler characteristic and the Hodge-Deligne polynomial.
