Polymers that start degrading under acidic conditions are increasingly investigated as a pathway to trigger the release of drugs once the drug carrier reached the slightly acidic tumour environment or after the drug carrier has been taken up by cells, resulting in the localization of the polymer in the acidic endosomes and lysosomes. The advances in the design of acid-degradable polymers and drug delivery systems have been summarized and discussed in this review article. Various acid-labile groups such as acetals, orthoester, hydrazones, imines and cis-aconityl, that can undergo cleavage in slightly acidic conditions, have been employed to create polymer architectures or polymer–drug conjugates that can degrade under lysosomal and endosomal conditions, triggering the fast release of drugs or DNA.