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Is space an entity that exists independently of matter (substantivalism), or is it only an abstraction from spatial relations between bodies (relationism)? Is there a lapse of time even when nothing changes, or is time only a measure of motion? Are motion and rest contrary properties or states of a body, or are there only changes in the positions of bodies relative to one another? Philosophers and physicists have disputed these questions from antiquity to the present day. We survey the arguments and attempt to find a resolution. But there are further questions. What is the significance of incongruent counterparts (left hands vs. right hands)? Is there a fact of the matter as to the geometry of space (flat, hyperbolic or elliptical), or as to whether space-like separated events occur at the same time? What is the principle of relativity? Does Einstein’s theory have consequences for the substantivalist/relationist debate? What is the status of spacetime in current physics and cosmology? Why does time but not space have a “direction”? Are past, present and future objective features of reality, or are they merely “stubborn illusions”? Does time flow? If not, how do we account for our sense of the passage of time?