The subject of antenna design, primarily a discipline within electrical en- neering, is devoted to the manipulation of structural elements of and/or the electrical currents present on a physical object capable of supporting such a current. Almost as soon as one begins to look at the subject, it becomes clear that there are interesting mathematical problems which need to be addressed, in the ?rst instance, simply for the accurate modelling of the electromagnetic ?elds produced by an antenna. The description of the electromagnetic ?elds depends on the physical structure and the background environment in which thedeviceistooperate. It is the coincidence of a class of practical engineering applications and theapplicationofsomeinterestingmathematicaloptimizationtechniquesthat is the motivation for the present book. For this reason, we have thought it worthwhile to collect some of the problems that have inspired our research in appliedmathematics,andtopresenttheminsuchawaythattheymayappeal to two di?erent audiences: mathematicians who are experts in the theory of mathematical optimization and who are interested in a less familiar and importantareaofapplication,andengineerswho,confrontedwithproblemsof increasing sophistication, are interested in seeing a systematic mathematical approach to problems of interest to them. We hope that we have found the right balance to be of interest to both audiences. It is a di?cult task. Our ability to produce these devices at all, most designed for a part- ular purpose, leads quite soon to a desire to optimize the design in various ways. The mathematical problems associated with attempts to optimize p- formance can become quite sophisticated even for simple physical structures.