Abstract: I will give an overview of some of the solution theory of nonlinear wave equations such as the wave map or Yang-Mills equations. Wave maps are wave-equations taking their values in manifolds and are also known as nonlinear sigma models. Even though many questions remain unanswered, we do have some understanding of how the geometry of the target influences the long-time existence of the solutions (via positive vs. negative curvature, and the existence of finite energy harmonic maps). Special emphasis will be placed on the notion of energy critical equations.