The current review is devoted to nanozymes, i.e., nanostructured artificial enzymes which mimic the catalytic properties of natural enzymes. Use of the term “nanozyme” in the literature as indicating an enzyme is not always justified. For example, it is used inappropriately for nanomaterials bound with electrodes that possess catalytic activity only when applying an electric potential. If the enzyme-like activity of such a material is not proven in solution (without applying the potential), such a catalyst should be named an “electronanocatalyst”, not a nanozyme. This paper presents a review of the classification of the nanozymes, their advantages vs. natural enzymes, and potential practical applications. Special attention is paid to nanozyme synthesis methods (hydrothermal and solvothermal, chemical reduction, sol-gel method, co-precipitation, polymerization/polycondensation, electrochemical deposition). The catalytic performance of nanozymes is characterized, a critical point of view on catalytic parameters of nanozymes described in scientific papers is presented and typical mistakes are analyzed. The central part of the review relates to characterization of nanozymes which mimic natural enzymes with analytical importance (“nanoperoxidase”, “nanooxidases”, “nanolaccase”) and their use in the construction of electro-chemical (bio)sensors (“nanosensors”).