How proteins manage to fold and how chaperones manage to assist the folding

•Reversible domain melting helps to solve the Levinthal's paradox of protein folding.•This allows estimating the spontaneous folding rate dependence on the domain size.•Chaperones provide the folding chain two ways to avoid inappropriate interactions. This review presents the current understanding of (i) spontaneous self-organization of spatial structures of protein molecules, and (ii) possible ways of chaperones’ assistance to this process. Specifically, we overview the most important features of spontaneous folding of proteins (mostly, of the single-domain water-soluble globular proteins): the choice of the unique protein structure among zillions of alternatives, the nucleation of the folding process, and phase transitions within protein molecules. We consider the main experimental facts on protein folding, both in vivo and in vitro, of both kinetic and thermodynamic nature. We discuss the famous Levinthal's paradox of protein folding and its solution, theoretical models of protein folding and unfolding, and the dependence of the rates of these processes on the protein chain length. Special attention is paid to relatively small, single-domain, and water-soluble globular proteins whose structure and folding are much better studied and understood than those of large proteins, especially membrane or fibrous proteins. Lastly, we describe the chaperone-assisted protein folding with an emphasis on the chaperones’ ability to prevent proteins from their irreversible aggregation. Since the possible assistance mechanisms connected with chaperones are still debatable, experimental data useful in selecting the most likely mechanisms of chaperone-assisted protein folding are presented.