I will give an overview of general relativity from the point of view of partial differential equations (PDEs). A crucial fact to establish about a given PDE system is whether or not it is well-posed, that is, whether or not unique solutions that depend continuously on given data exist. I will focus in particular on the gauge freedom of general relativity and the freedom it brings to the formulation of the initial value problem. Well-posedness of the initial value problem depends on whether or not the PDE system is in some sense wave-like. I will describe how the wave-like nature of gauge choices can be characterized, and how they may be coupled to the Einstein field equations. Finally I will consider the question: what are the set of gauge conditions that may be coupled to general relativity to form a wave-like PDE system?