The present invention is directed to branched reactive water-soluble polymers comprising at least two polymer arms, such as poly(ethylene glycol), attached to a central aliphatic hydrocarbon core molecule through heteroatom linkages. The branched polymers bear at least one functional group for reacting with a biologically active agent to form a biologically active conjugate. The functional group of the branched polymer can be directly attached to the aliphatic hydrocarbon core or via an intervening linkage, such as a heteroatom, -alkylene-, —O-alkylene-O—, -alkylene-O-alkylene-, -aryl-O—, —O-aryl-, (—O-alkylene-)_m, or (-alkylene-O—)_m linkage, wherein m is 1-10.