Even since Kummer's 19th century attempts to prove Fermat's Last Theorem, cyclotomic fields have been at the center of attention. Indeed, in the second half of the 20th century the subject enjoyed a strong revival in the hands of Iwasawa. Despite tremendous progress, many questions remain. Vandiver's conjecture is perhaps the most outstanding one. In this talk we discuss the problem of calculating class numbers of cyclotomic fields. This is a computational problems that, even using the fastest computers, has only been solved in a handful of cases. We present an experimental approach to determining these class numbers.