High-vacuum mechanical pumps are widely used in various fields of science and technology for research and industrial applications. Modern high-vacuum mechanical pumps include molecular vacuum pumps, turbomolecular vacuum pumps and hybrid turbomolecular vacuum pumps, i.e., designs that represent a turbomolecular vacuum pump with one or more additional molecular stages. There are also a number of experimental and theoretical developments that propose combinations of different types of high-vacuum pumps in which, as a rule, on the suction side the main pumping is carried out by an axial turbomolecular stage. These schemes belong to the class of modern high-vacuum mechanical pumps. The paper examines the designs and analyzes scientific studies of modern high-vacuum mechanical pumps. Relying on this overview, we found the unsolved problems of designing and mathematical modeling of high-vacuum mechanical pumps working processes, which could give further prospects for vacuum technology development