The subject of this talk is wave equations that arise from geometric considerations. Prime examples include the wave map equation and the Yang-Mills equation on the Minkowski space. On one hand, these are fundamental field theories arising in physics; on the other hand, they may be thought of as the hyperbolic analogues of the harmonic map and the elliptic Yang-Mills equations, which are interesting geometric PDEs on their own.Our main concern will be global well-posedness for large data of these PDEs in dimensions where the conserved energy is critical with respect to the scaling symmetry of the equations. I will first explain the ‘threshold conjecture’ for wave maps and its resolution by Sterbenz-Tataru (see also the work by Krieger-Schlag and Tao), as well as its latest refinement in my work with A. Lawrie. I will also describe my recent work with D. Tataru on the global well-posedness of the energy critical Maxwell-Klein-Gordon system, which shares many similarities with the Yang-Mills equation.