We present a theory of discontinuous motion of particles in continuous space-time. We show that the simplest non-relativistic evolution equation for such motion is just the Schr?dinger equation in quantum mechanics. This strongly implies that what quantum mechanics describes is discontinuous particle motion. Considering the fact that space-time may be essentially discrete when considering gravity, we further present a theory of discontinuous motion of particles in discrete space-time. We show that the evolution of discontinuous motion may naturally result in the dynamical process of wave-function collapse, and that this collapse will bring about the appearance of continuous motion of objects in the macroscopic world.