In this paper I challenge the usual interpretations of Newton's and Leibniz's views on the nature of space and the relativity of motion. Newton's 'relative space' is not a reference frame: and Leibniz did not regard space as defined with respect to actual enduring bodies. Newton did not subscribe to the relativity of inertial motions; whereas Leibniz believed no body to be at rest, and Newton's absolute motion to be a useful fiction. A more accurate rendering of the opposition between them, I argue, leads to a wholly different understanding of Leibniz's theory of space, one which is not susceptible to the objections Newton had raised against Descartes regarding the representation of motion. This in turn suggests a new approach for contemporary theory of space, one which neither hypostatizes space nor tries to reduce it to relations among actual things, 1 The Opposition between Newtoll and Leibniz 2 Newton Oil Space and Relativity 3 Newton alld the Structure of Space 4 Relative Space alltl Absolute Motion in Leibniz 5 I.eibniz and tile Construction of Space 6 Conclusion.