When a fluid filled container is shaken vertically, one may observe waves on the surface of that container if the shaking is sufficiently strong. These waves arise out of a subharmonic instability: they have half the frequency of the shaking, and are called Faraday waves. In separate experiments, high-speed films of droplet impacts on static fluid baths show that the droplet does not always coalesce with the bath on impact, but may bounce a few times before coalescing. Combining these two experimental facts, about 10 years ago it was discovered that millimetric liquid droplets can bounce indefinitely when dropped on the surface of a bath of the same liquid in a shaken container. The phenomenon occurs below the shaking threshold for the Faraday instability. More surprisingly, the droplets can also spontaneously "walk" along the surface of the vibrating bath. These walkers then exhibit many features previously thought to be exclusive to quantum mechanics such as wave-particle duality, quantised energy states, single particle diffraction and tunnelling behaviour. The aim of this proposal is to explore the fluid mechanical aspects of this system. There are many unanswered challenging questions due to the complexity of the problem. Fluid mechanics questions include the understanding of non coalescing drop impact, and the behaviour of reflecting walkers and their pilot wave field at walls. We will also seek to understand how a purely classical mechanics system exhibits quantum mechanical-like behaviour, and probe the limits of this analogy. The proposed research involves a combination of analytical and numerical approaches as well as comparisons with experiments. We will partner with an MIT state-of-the-art fluid dynamics laboratory which will provide both experimental data and design validation experiments. The problems we plan to study are of general interest in fluid mechanics and in the theory of free boundary problems and dynamical systems. It is expected that the results will have broad applications, in particular to the understanding of the impact of drops and particles with fluids. Faraday instabilities are also the most reliable way of generating consistently sized droplets continuously. Because of this there are several possible microfluidics applications for this research, such as developing better devices for delivering inhaled drugs.