  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-Tc materials such as
Nb$_3$Sn, NbN, and MgB$_2$. Indeed, beyond the so-called superheating field
$H_{\mathcal{sh}}$, flux will spontaneously penetrate even a perfect
superconducting surface and ruin the performance. We present intuitive
arguments and simple estimates for $H_{\mathcal{sh}}$, 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 MgB$_2$? 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.
