Universally meager sets

We study category counterparts of the notion of a universal measure zero se t of reals. We say that a set A subset of or equal to R is universally meager if every Borel isomorphic image of A is meager in R. We give various equivalent defi nitions emphasizing analogies with the universally null sets of reals. In particular, two problems emerging from an earlier work of Grzegorek are solved.