iwasawa theory concerns the growth of arithmetic objects in towers of number fields. the unramified iwasawa modules that are the inverse limits of p-parts of class groups up such a tower are central objects in the study of iwasawa theory. i will discuss a variety of conjectures and results on the structure of unramified iwasawa modules, along with applications. in particular, i intend to explain iwasawa-theoretic refinements of leopoldt's reflection principle which form part of joint work with bleher, chinburg, greenberg, kakde, pappas, and taylor.