This paper deals with discrete computational geometry of motion. It combines concepts from the fields of kinematics and computer aided geometric design and develops a computational geometric framework for geometric construction of motions useful in mechanical systems animation, robot trajectory planning and key framing in computer graphics. In particular, screw motion interpolants are used in conjunction with deCasteljau-type methods to construct Bézier motions. The properties of the resulting Bézier motions are studied and it is shown that the Bézier motions obtained by application of the deCasteljau construction are not, in general, of polynomial type and do not possess the useful subdivision property of Bernstein-Bézier curves. An alternative from of deCasteljau algorithm is presented that results in Bézier motions with subdivision property and Bernstein basis function. The results are illustrated by examples.