Let
 ,
 ,
 and
 denote respectively the variety of groups of
exponent dividing
 , the variety of nilpotent groups of class at
most
 , the class of nilpotent groups and the class of
finite groups. It follows from a result due to
Kargapolov and Curkin and independently to Groves
that in a variety not containing all metabelian
groups, each polycyclic group
 belongs to
 . We show that
 is in fact in
 , where
 is an integer depending only on the variety. On
the other hand, it is not always possible to find
an integer
 (depending only on the variety) such that
 belongs to
 , but we
characterize the varieties in which that
is possible. In this case, there exists a
function
 such that, if
 is
 -generated, then
 . So, when
 , we obtain an extension of Zel'manov's result
about the restricted Burnside problem (as one
might expect, this result is used in our proof).
Finally, we show that the class of locally
nilpotent groups of a variety
 forms a variety if and only if
 for some integers
 .
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