Theoretical limits to the performance of superconductors in high magnetic fields parallel to their surfaces are of key relevance to current and future accelerating cavities, especially those made of new higher-T c materials such as Nb3Sn, NbN, and MgB2. Indeed, beyond the so-called superheating field , flux will spontaneously penetrate even a perfect superconducting surface and ruin the performance. We present intuitive arguments and simple estimates for , and combine them with our previous rigorous calculations, which we summarize. We briefly discuss experimental measurements of the superheating field, comparing to our estimates. We explore the effects of materials anisotropy and the danger of disorder in nucleating vortex entry. Will we need to control surface orientation in the layered compound MgB2? Can we estimate theoretically whether dirt and defects make these new materials fundamentally more challenging to optimize than niobium? Finally, we discuss and analyze recent proposals to use thin superconducting layers or laminates to enhance the performance of superconducting cavities. Flux entering a laminate can lead to so-called pancake vortices; we consider the physics of the dislocation motion and potential re-annihilation or stabilization of these vortices after their entry. © 2017 IOP Publishing Ltd.