Polymer nanoparticles and nano/micromotors are novel nanostructures that are of increased interest especially in the diagnosis and therapy of cancer. These structures are modified by antibodies or nucleic acid aptamers and can recognize the cancer markers at the membrane of the cancer cells or in the intracellular side. They can serve as a cargo for targeted transport of drugs or nucleic acids in chemo- immuno- or gene therapy. The various mechanisms, such as enzyme, ultrasound, magnetic, electrical, or light, served as a driving force for nano/micromotors, allowing their transport into the cells. This review is focused on the recent achievements in the development of polymer nanoparticles and nano/micromotors modified by antibodies and nucleic acid aptamers. The methods of preparation of polymer nanoparticles, their structure and properties are provided together with those for synthesis and the application of nano/micromotors. The various mechanisms of the driving of nano/micromotors such as chemical, light, ultrasound, electric and magnetic fields are explained. The targeting drug delivery is based on the modification of nanostructures by receptors such as nucleic acid aptamers and antibodies. Special focus is therefore on the method of selection aptamers for recognition cancer markers as well as on the comparison of the properties of nucleic acid aptamers and antibodies. The methods of immobilization of aptamers at the nanoparticles and nano/micromotors are provided. Examples of applications of polymer nanoparticles and nano/micromotors in targeted delivery and in controlled drug release are presented. The future perspectives of biomimetic nanostructures in personalized nanomedicine are also discussed.