Cancer stem cells (CSCs) are a great challenge in the fight against cancer because these self-renewing tumorigenic cell fractions are thought to be responsible for metastasis dissemination and cases of tumor recurrence. In comparison with non-stem cancer cells, CSCs are known to be more resistant to chemotherapy, radiotherapy, and immunotherapy. Elucidation of mechanisms and factors that promote the emergence and existence of CSCs and their high resistance to cytotoxic treatments would help to develop effective CSC-targeting therapeutics. The present review is dedicated to the implication of molecular chaperones (protein regulators of polypeptide chain folding) in both the formation/maintenance of the CSC phenotype and cytoprotective machinery allowing CSCs to survive after drug or radiation exposure and evade immune attack. The major cellular chaperones, namely heat shock proteins (HSP90, HSP70, HSP40, HSP27), glucose-regulated proteins (GRP94, GRP78, GRP75), tumor necrosis factor receptor-associated protein 1 (TRAP1), peptidyl-prolyl isomerases, protein disulfide isomerases, calreticulin, and also a transcription heat shock factor 1 (HSF1) initiating HSP gene expression are here considered as determinants of the cancer cell stemness and potential targets for a therapeutic attack on CSCs. Various approaches and agents are discussed that may be used for inhibiting the chaperone-dependent development/manifestations of cancer cell stemness.