The invention concerns novel cationic polymers of formula (I) in which n is a whole number between 0 and 5 and p is a whole number between 2 and 20,000, more particularly p ranges between 10 and 18,000, and advantageously between 200 and 1,000, characterized in that: at least 10%, advantageously 30 to 80%, preferably 70%, of the free NH.sub.2 functions are substituted by identical or different hydrophilic R groups; said cationic polymer can further comprise at least a targeting element associated covalently or not with the free NH.sub.2 functions and/or said hydrophilic R groups provided that said cationic: polymer contains at least 20%, preferably at least 30%, of free NH.sub.2 functions. The invention also concerns a complex comprising at least said cationic polymer associated with at least a therapeutically active substance, in particular a nucleic acid comprising at least a negative charge. The invention is useful for transferring a nucleic acid into target cells, in particular for gene therapy.