 What is the nature of motion in physical theories and theorising, and is there any significance to the distinction between ‘absolute’ and ‘relative’ motion? In the companion article, on  absolute and relational space and motion: classical theories,  we discussed how such questions were addressed in the history of physics from Aristotle through to Newton and Leibniz. In this article, we explore the ways in which the selfsame issues have been taken up by contemporary authors, beginning with Mach, moving on to Einstein, and concluding with a discussion of two highly relevant modern research programmes: shape dynamics and the so-called ‘dynamical approach’ to spacetime. Readers interested in following up either the historical or the current debates about the natures of space, time and motion will find ample links and references scattered through the discussion and in the  Other Internet Resources  section below.