The invention provides a water-soluble polymer comprising (i) an imide group comprising a linking nitrogen atom, a first carbonyl group covalently attached to the linking nitrogen atom, and a second carbonyl group covalently attached to the linking nitrogen atom; (ii) a first water-soluble polymer segment covalently attached, either directly or through one or more atoms, to the first carbonyl group of the imide group and (iii) a second water-soluble polymer segment covalently attached, either directly or through one or more atoms, to the second carbonyl group of the imide group. The invention also provides, among other things, methods for preparing polymers, conjugates, pharmaceutical compositions and the like.