Members of the HSP70/HSP110 family (HSP70s) form a central hub of the chaperone network controlling all aspects of proteostasis in bacteria and the ATP-containing compartments of eukaryotic cells. The heat-inducible form HSP70 (HSPA1A) and its major cognates, cytosolic HSC70 (HSPA8), endoplasmic reticulum BIP (HSPA5), mitochondrial mHSP70 (HSPA9) and related HSP110s (HSPHs), contribute about 3% of the total protein mass of human cells. The HSP70s carry out a plethora of housekeeping cellular functions, such as assisting proper de novo folding, assembly and disassembly of protein complexes, pulling polypeptides out of the ribosome and across membrane pores, activating and inactivating signaling proteins and controlling their degradation. The HSP70s can induce structural changes in alternatively folded protein conformers, such as clathrin cages, hormone receptors and transcription factors, thereby regulating vesicular trafficking, hormone signaling and cell differentiation in development and cancer. To carry so diverse cellular housekeeping and stress-related functions, the HSP70s act as ATP-fuelled unfolding nanomachines capable of switching polypeptides between different folded states. During stress, the HSP70s can bind (hold) and prevent the aggregation of misfolding proteins and thereafter act alone or in collaboration with other unfolding chaperones to solubilize protein aggregates. Here, we discuss the common ATP-dependent mechanisms of holding, unfolding-by-clamping and unfolding-by-entropic pulling, by which the HSP70s can apparently convert various alternatively folded and misfolded polypeptides into differently active conformers. Understanding how HSP70s can prevent the formation of cytotoxic protein aggregates, pull, unfold, and solubilize them into harmless species is central to the design of therapies against protein conformational diseases.