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Superconducting radio frequency (SRF) cavities used in present-day accelerators for the acceleration of charged particles near the speed of light are based on the axially symmetric TM010 mode of a pillbox cavity. Future accelerators such as the Linear Collider require high accelerating gradients to limit the length of the linac. Two techniques to improve the gradient are being explored: a cavity that is half reentrant to improve the electromagnetic characteristics, and improved heat transfer via cooling channels and surface modification at the helium interface. These changes could potentially increase the gradients and reduce the cryogenic losses. For other applications more important criteria are simplicity, acceleration of high beam current, or the ability to use advanced materials such as Nb3Sn or high-Tc superconductors. A new type of cavity based on the TM01p pillbox mode with p>0 offers such improvements.