  We say that a group $G$ has Bergman's property (the property of universality
of finite width) if for every generating set $X$ of $G$ with $X=X^{-1}$ we have
that $G=X^k$ for some natural number $k.$ The property is named after George
Bergman who have proved recently that the infinite symmetric groups are groups
of universally finite width.
  The first example of an infinite group with Bergman's property is due to
Shelah (1980s). Lately some other examples have been found: the automorphism
groups of doubly transitive chains (Droste-Holland), the automorphism group of
reals as a Borel space (Droste-G\"obel) and the infinite-dimensional general
linear groups over division rings.
  In the present paper we prove that the automorphism group $Aut(N)$ of any
infinitely generated free nilpotent group $N$ has Bergman's property, is
generated by involutions, perfect and has confinality greater than $rank(N).$
Also, we obtain a partial answer to a question posed by Bergman establishing
that the automorphism group of a free group of countably infinite rank is a
group of universally finite width.
