Drug delivery via block copolymer nanoassemblies has been widely viewed as an efficient way to augment therapeutic efficiency of anticancer drugs for cancer therapy. Acid-degradable block copolymer-based nanoassemblies are promising intracellular candidates for tumor-targeting drug delivery as they exhibit enhanced release of encapsulated drugs through their dissociation (or disintegration) upon cleavage of acid-labile linkages in response to a tumor's acidic environment (pH = 4.2–6.7). Furthermore, degradation of the self-assemblies via acid-catalyzed hydrolysis can be tuned with a choice of various available acid-labile linkages. This review summarizes recent advances in the synthesis and disassembly of acid-cleavable block copolymers for controlled drug delivery. Strategies to integrate acid-labile covalent linkages in block copolymers at different locations, as in micelle cores and core/corona interfaces are particularly focused upon. Additionally, elegant strategies that allow for the synthesis of dual acid/reduction, acid/light, acid/enzyme-degradable block copolymer systems with cleavable linkages in single or dual locations are discussed.