Spacetime and quantum mechanics are the pillars of our modern understanding of fundamental physics. But there are storm clouds on the horizon indicating that these principles are approximate, and must be replaced with something deeper. The union of quantum mechanics and gravity strongly suggests that spacetime as a basic concept is doomed, and there are related indications of fundamental limitations to quantum mechanics in both the early and late universe. In this talk I review these paradoxes and describe indications for a new picture where spacetime and quantum mechanics will be seen to emerge hand in hand from more primitive principles, making contact with new areas of mathematics. I give concrete examples of how these ideas work in the context of scattering amplitudes, describing particle collision experiments of the sort performed at the Large Hadron Collider.