Overrunning clutches are devices for transmitting rotary motion in one direction only. These mechanisms are widely used in automotive industry, for example, in torque converters, impulse stepless transmissions, inertial continuously variable transmissions, starter engine starting system, and in other similar devices, where torque transmission is performed only in one direction. There are many different designs of the overrunning clutches, for example, ball, roller, cam, ratchet, spring ones, etc. But despite such a variety of designs and great efforts to establish reliable overrunning clutches, these mechanisms are still the weakest parts of many drive systems. Therefore, creation of reliable overrunning clutches is an urgent problem of mechanical engineering. Unfortunately, existing designs of the overrunning clutches have insufficient reliability and durability, which in many cases limits reliability of drive as a whole. The weakest links of the overrunning clutches are the so called wedging elements.This paper describes a promising new design of overrunning clutches. In this design only a small part of torque is transmitted through the weak wedging elements, and the main part of this torque is transmitted through friction disc surfaces, which allows to unload the wedging elements and substantially improve the reliability and durability of the overrunning clutches in comparison with known designs of overrunning clutches. Investigation of the redistribution of amount of torque transmitted through the wedging elements and the friction disc surfaces was done. It was shown that in suggested design there is the principal possibility of reducing of the amount of the torque transmitted through the wedging elements in tens and hundreds times. Besides, it was developed and investigated a mathematical model describing the dynamics of the overrunning clutches of relay type. Feature of the developed model is that, despite of the variability of the structure, dynamics of the overrunning clutches is described by only one system of differential equations, which greatly simplifies the study of periodic solutions and their stability.