Abstract: Iwasawa theory is the study of the growth of arithmetic invariants in Galois extensions of global fields with Galois group a p-adic Lie group. Beginning with Iwasawa's seminal work in which he proved that the p-primary part of the class group in Z_p-extensions of number fields grows with striking and unexpected regularity, Iwasawa theory has become a central strand of modern number theory and arithmetic geometry. While the theory has traditionally focused on towers of number fields, the function field setting has been studied extensively, and has important applications to the theory of p-adic modular forms. This talk will introduce an exciting new kind of p-adic Iwasawa theory for towers of function fields over finite fields of characteristic p, and discuss some applications to problems and open conjectures in the field.