Iwasawa Theory for elliptic curves/modular forms has been traditionally in better shape at ordinary primes than at supersingular ones. After sketching the ordinary theory, we will indicate what makes the supersingular case more complicated, and then introduce *pairs* of objects that that are as simple as their ordinary counterparts. These pairs of objects work in tandem to shed some light on the nature of ranks of elliptic curves and the size of Sha along cyclotomic Z_p extensions.