Classical Iwasawa theory has to do with the rate of growth of ideal class groups in towers of number fields. In this talk I'll survey some of the history of the subject. This includes various Main Conjectures which link the above growth rates to analytically defined invariants such as p-adic L-series. By the end of the talk I'll describe how previous Main Conjectures are about first Chern classes. The theory of higher Chern classes suggests a new direction for the subject.