(57) Polymers containing polyalkylpiperidines and use thereof as stabilizers for synthetic polymer compositions are described. In said polymers, sterically hindered polyalkylpiperidines are linked in the main chain via bridging members containing 2-hydroxy-1,3-trimethylene groups, or derivatives thereof. The polymers are distinguished by an improved stabilizing effect in various synthetic polymer compositions, such as polyolefins, against light- and/or heat-induced deterioration thereof. The polyaikylpiperidine-containing polymers of the invention are hardly volatile upon heat- processing or during storage of shaped articles containing said polymers, and are resistant to extraction with solvents.