Some of the most fascinating pieces of mathematics, such as Dirichlet's class number formula and the celebrated Birch and Swinnerton-Dyer conjecture, build a bridge between the distant worlds of arithmetic and analysis. Euler systems and Iwasawa theory provide an intermediate step between the two, and both have been at the source of much of the progress to date on BSD conjecture and its many generalizations. In this talk, I will expand on some of these ideas, including a brief discussion of some of the recent developments in the area.