In Iwasawa theory, one attempts to relate arithmetic objects arising from Galois cohomology groups to analytic objects, specifically $p$-adic $L$-functions, usually at the level of a cyclotomic $Z_p$-extension of a number field. For some time, I have been interested in operations in the Galois cohomology of number fields with restricted ramification, particularly cup products. I will explain various applications of these operations, for instance to Iwasawa theory over certain nonabelian $p$-adic Lie extensions of number fields. I will then discuss the broad outline of a conjecture relating an inverse limit of such cup products up the cyclotomic tower to the two-variable $p$-adic $L$-function for Hida families of cusp forms congruent to Eisenstein series modulo $p$.