Yves Couder and coworkers have recently discovered that droplets walking on a vibrating fluid bath exhibit several features previously thought to be peculiar to the microscopic, quantum realm. These walking droplets propel themselves by virtue of a resonant interaction with their own wave field. Thus, the system is reminiscent of the double-wave pilot-wave dynamics envisaged by Louis de Broglie. New theoretical developments provide rationale for the complex behavior of the bouncing droplets, and yield a trajectory equation for the walking droplets. Experimental and theoretical results in turn reveal and rationalize the emergence of quantization and wave-like statistics from pilot-wave dynamics in a number of settings.