The engagement of gearwheels with curvilinear teeth is intended to create small-sized mechanical gears of rotational motion with big gear ratio in one stage. The smaller gearwheel—pinion (16)—has one tooth, having in its face section the shape of circumference (3), eccentrically shifted with respect to the axis OO1 of the gearwheel (16). The curvilinear helical tooth of the gearwheel (16) (helical eccentric) is generated by sequential shifting of the circumference 3 along the axis OO1 and its continuous turning around the axis. The greater gearwheel (17) has helical teeth, generated by turning of the cycloidal curve (5), the teeth are conjugated with the helical surface of the pinion (16). The engagement has a continuous line of contact along the whole length of the tooth, where in each section a circular pin tooth and a cycloid are engaged, having minimum losses for friction. In order to eliminate axial loads, occurring in engagement of helical teeth, the gearwheels (16) and (17) are made herring-bone. The similar eccentrically cycloidal engagement can be implemented by assembled gearwheels. On the basis of the engagement one can design cylindrical gearboxes with parallel shafts, bevel gearboxes with intersecting shafts, and also planetary gears according to David and James schemes, possessing the increased load carrying capacity at smaller overall dimensions.