The motion of fluids can be complicated, as we know whenever we see waves break on a beach, fly in an airplane, or look at a lake on a windy day. Euler in the 1750s proposed a mathematical model for incompressible fluids, and since then an immense amount of progress has been made. But huge problems are still beyond our reach. Surface water waves refer to the situation where the water lies below a body of air. Describing what we may see or feel at the beach or in a boat, water waves are a perfect specimen of applied mathematics. They host a wealth of phenomena ranging in length scale from ripples driven by surface tension to tsunamis and to rogue waves. This paper will review mathematical aspects of water wave phenomena, specifically, (1) is there a unique solution to the Cauchy problem? for how long a time does it exist? (2) are the solutions regular or do singularities develop after some time? (3) are there solutions spatially periodic? are they dynamically stable?