Article submitted to IET Microwaves, Antennas and Propagation. "Maximum bandwidth performance bounds are given for any finite antenna, either in free space or above a perfectly conducting ground plane. The theory is valid for any class of antenna, e.g. for patch antennas, dielectric resonator antennas, etc. whose radiating currents can be confined to the Chu sphere or hemisphere of radius a. An example is given using the equivalentcircuit of a perfect loop antenna and a full-wave numerical analysis of a realisable loop antenna."