  We obtain a formula for the generating series of (the push-forward under the
Hilbert-Chow morphism of) the Hirzebruch homology characteristic classes of the
Hilbert schemes of points for a smooth quasi-projective variety of arbitrary
pure dimension. This result is based on a geometric construction of a motivic
exponentiation generalizing the notion of motivic power structure, as well as
on a formula for the generating series of the Hirzebruch homology
characteristic classes of symmetric products. We apply the same methods for the
calculation of generating series formulae for the Hirzebruch classes of the
push-forwards of "virtual motives" of Hilbert schemes of a threefold. As
corollaries, we obtain counterparts for the MacPherson (and Aluffi) Chern
classes of Hilbert schemes of a smooth quasi-projective variety (resp. for
threefolds). For a projective Calabi-Yau threefold, the latter yields a Chern
class version of the dimension zero MNOP conjecture.
