Multibody mechanical systems (mechanisms and machine drives) are widely used in different fields of modern engineering due to their reliability and simple design. They can be found in robots, manipulators, technological and construction equipment, automatic lines, etc. This paper presents a unified theory of structure, synthesis and analysis of mechanisms and machines with geometrical (single and multiple kinematic pairs), flexible contact (friction or belt) and dynamic contactless (inertial, gravitational, etc.) connections. The theory can be used to construct planar and spatial single- and multi-loop kinematic chains of machines with a given number of closed loops and driving motors. Areas of possible existence of multibody mechanical systems with open, closed and mixed kinematic chain are determined. Based on these findings, various planar and spatial gear and linkage patentable mechanisms are developed that can be used in vibrational drives, variable structure systems requiring precise stoppage during the cycle, lever actuators of multi-axle locomotives, spatial mixers with several mixing tanks, tribometers for measuring the limiting pulling capacity of flexible belts of belt-and-pulley drives, and direct-drive devices for horizontal motion of a suspended load with a low set velocity.