I will report on my recent joint work with Greither on the Galois module structure of $\ell$-adic realizations of Picard $1$-motives in positive characteristic and their Iwasawa theoretic analogues in characteristic 0. In the process, I will state and sketch the proof of an equivariant Main Conjecture in Iwasawa theory. Finally, I will discuss applications of the equivariant Main Conjecture ranging from proofs of refinements of the Brumer-Stark and Coates-Sinnott Conjectures to proofs of particular cases of the Equivariant Tamagawa Number Conjecture of Bloch-Kato and the non-abelian Main Conjecture of Coates-Fukaya-Kato-Sujatha-Venjakob. This is a two hour lecture, the first half emphasizing the more geometric aspects of our work, while the second will focus on the number theoretic aspects and applications. Talk time runs until 4:00 PM.