We give a brief review of the current status of world research in the field of creating quantum sensors of DC and RF electric fields based on highly excited Rydberg atoms with principal quantum number n ≫ 1. Such atoms have large dipole moments, which increase as n2 with increasing n. Due to this fact, electric polarizabilities of Rydberg levels increase as n7 and exceed the polarizabilities of low-excited atoms by many orders of magnitude. This forms the basis for creating highly sensitive quantum sensors. For their implementation, alkali-metal atoms in compact optical vapor cells are excited to Rydberg states. When microwave radiation with a frequency above 1 GHz should be detected, splitting of a single resonance of electromagnetically induced transparency (EIT), which appears under two-photon laser excitation of Rydberg states, into two resonances due to the Autler-Townes (AT) effect in a microwave radiation field is measured. With Rydberg sensors, it is possible to construct both metrological standards for measuring electric field strength and highly sensitive detectors of RF fields for various applications. Our first experimental results on the observation of EIT resonances under two-photon laser excitation 5S1/2 → 5P3/2 → nS1/2 of 85Rb Rydberg states in an optical cell and on the observation of AT splitting in the field of microwave radiation with a frequency of 58.17 GHz, which was in resonance with 41S1/2 → 41P3/2 transition between the neighboring Rydberg states, are also presented.