Optical tweezers and acoustic traps use forces exerted by waves to confine small objects in three dimensions and to transport them along pre-programmed trajectories. This is not the end of the story, however, because trapped objects scatter the waves that trap them, and the scattered waves mediate interactions among the objects that give rise to collective behavior that is not (explicitly) programmed into the traps. Some such wave-matter composite systems settle into well-defined states of organization whose behavior can be understood within the framework of the recently developed theories of photokinetic and autokinetic effects. We will introduce wave-matter composite systems with canonical examples such as the wave-driven oscillator and the non-reciprocal Einstein solid. Other examples, such as acoustic chaining of fluid droplets, will be presented as outstanding challenges. If all goes well, some aspects of wave-matter composite behavior will be illustrated with practical demonstrations.