Abstract: Oily hyperbranched polymers derived from ethylene, propylene, butene and/or a C5-C24 &agr;-olefin, and a method for their synthesis, are disclosed. The polymers have non-regular microstructures and are characterized by a ratio ()of methyl hydrogens centered around 0.85 ppm on the 1H-NMR spectra of the polymers relative to total aliphatic hydrogens of from about 0.40 to about 0.65 for polymers derived from ethylene or butene, and a ratio ()of from greater than 0.50 to about 0.65 for polymers derived from propylene. A method for grafting hyperbranched polymers derived from ethylene, propylene, butene and/or a C5-C24 &agr;-olefin onto aromatic rings in organic molecules and polymers, and the resulting grafted materials, are also disclosed. The hyperbranched polymers and grafted materials are useful, for example, as lubricants and lubricant additives.