I survey briefly the main lessons about the nature of physical time from (special
and general) relativistic physics. They amount to a progressive dismissal of
commonsensical assumptions about time and a difficult learning process about what
are its real physical features. I then outline how the search for a quantum theory
of gravity and spacetime is challenging, even more, our understanding of the
spatiotemporal aspects of the world, and suggests a more radical, complete
disappearance of time (and space) from the fundamental structure of the universe.
The conceptual challenges of this radical disappearance of time will be
emphasized, alongside the physical ones.