This paper presents the concept of variable radius drum mechanisms (VRDMs). A drum, or spool, consists of a spindle with flanges, around which a cable is wound. The cylindrical surface of an ordinary spool has a constant radius. In a variable radius drum (VRD), the radius of the spool varies along its profile. Properties of such devices are discussed, as well as the kinematic analysis and synthesis. The main contribution of the work is the theory of the VRD synthesis problem, rooted in a closed-form analytical solution. In order to highlight the benefits of VRDMs, two applications are presented and analyzed as examples. The first example consists of a mechanism which can support and guide a load along a horizontal linear path. The second example shows a solution to improve the locomotion of a legged robot. Finally, a prototype is made on the basis of the first case scenario and its performance is evaluated and discussed, showing a remarkable accuracy, with a deviation from the nominal trajectory of less than 1%.