  A power structure over a ring is a method to give sense to expressions of the
form $(1+a_1t+a_2t^2+\ldots)^m$, where $a_i$, $i=1, 2,\ldots$, and $m$ are
elements of the ring. The (natural) power structure over the Grothendieck ring
of complex quasi-projective varieties appeared to be useful for a number of
applications. We discuss new examples of $\lambda$- and power structures over
some Grothendieck rings of varieties. The main example is for the Grothendieck
ring of maps of complex quasi-projective varieties. We describe two natural
$\lambda$-structures on it which lead to the same power structure. We show that
this power structure is effective. In the terms of this power structure we
write some equations containing classes of Hilbert-Chow morphisms. We describe
some generalizations of this construction for maps of varieties with some
additional structures.
