With the development of nanotechnology, new scientific directions have been initiated and a significant amount of researches has been carried out, dedicated to the creation and search for promising applications in medicine, in particular, oncology, of nanocomposites based on bioinert, biocompatible and bioactive nanoparticle materials and modern chemotherapeutic drugs with different mechanisms of action. These works contain data indicating the advantages to introduce nanocomposite drugs into clinical practice, compared to the traditional use of chemotherapeutic drugs. The research results confirm the priority of works in the field of creation of new nanocomposite chemotherapeutic drugs for use in antitumor therapy and overcoming of drug resistance of malignant cells and neoplasms. The purpose of the review is to generalize and analyze the authors’ experimental works performed at Chuiko Institute of Surface Chemistry of National Academy of Sciences of Ukraine, concerning the specifics of the effect of chemotherapeutic drugs with different mechanisms of action and related nanocomposites, on cellular systems and tumors. Such data are relevant to determine promising directions and ways of creation of new effective nanocomposite drugs for use, in particular, in antitumor chemotherapy. At Chuiko Institute of Surface Chemistry of National Academy of Sciences of Ukraine, using the chemical engineering method, magnetically sensitive core-shell nanocomposites were synthesized with a multi-level hierarchical nanoarchitecture capable of performing a complex of diagnostic and therapeutic functions with various mechanisms of action, characteristic of medico-biological nanorobots. Their use makes it possible to implement the principle of targeted delivery of antitumor drugs, to carry out real-time local therapy by various methods and MRI diagnosis of the disease, to realize the synergy of treatment with antitumor drugs of different mechanisms of action, to reduce the toxic side effects of oncological drugs on the body. The combination of the above features of nanocomposites, as well as their role in overcoming the drug resistance of cells of malignant neoplasms to cisplatin, testify to the perspective of creation and research of nanocomposite drugs in solving the problem of increase in the efficiency of chemotherapy.