Geometry and kinematics have been intimately connected in their historical development, and the exploitation of this connection is key to contemporary applications in computer-aided design, robotics, manufacturing, and related areas. In this survey, we explore a variety of classical and modern problems that illustrate how simple rules of motion generate interesting curves and, conversely, the computational problems that arise in producing motions with prescribed paths and speeds. These encompass the geometry of trajectories under centripetal forces; the use of mechanisms to transform rotary motion into motion along general curves; real-time interpolators for digital motion control; and the specification of spatial motions that involve continuous variation of position and orientation. Such examples illustrate both the intellectual appeal and practical benefits of a sustained synergy between dialog between the study of curves and of motions.