The plane-parallel rolling of a disc along a fairly smooth curve under the action of perturbing and controlling forces and moments of forces is investigated. Models of the controlled motion corresponding to an explicit or implicit coordinate specification of the curve are constructed. The requirements imposed on the values of the sliding friction force and the normal pressure force, which ensure that the disc rolls without slipping and without loss of contact are determined. The problem of bringing the disc into the required state of motion in a time-optimal way by the action of the force of gravity and a limited axial moment of forces are considered. The situation when the reference curve contains a linear component (inclined upwards or downwards) and a periodic (wavy) component is investigated. Problems of optimal synthesis for a relatively large and relatively small controlling moment are considered.Investigations of the rolling of bodies of different geometrical shape on surfaces, both flat and curved (see Refs 1-4, etc.), are important for applications in modern machine construction, transport, mobile robot systems, 5-7 etc. However, there are no results of systematic investigations of the problems of the dynamics and control of rolling of bodies on curved surfaces, including comparatively simple bodies (cylinders, discs and spheres). From the applied point of view, the problem of controlled rolling of a disc under gravitational forces on a curve containing a linear incline and periodic smooth changes in the slope is of interest as a model of the motion of a wheel along a path with a curvilinear profile. The solution of problems of the control of the motion of a disc along a closed curve, for example, inside or outside a circle, surmounting obstacles or "rolling out" from hollows of different shapes etc. is of fundamental importance.