A user's guide to Beilinson-Kato's zeta elements

In his ground-breaking work, K. Kato constructed the Euler system of Beilinson--Kato's zeta elements and proved spectacular results on the Iwasawa main conjecture for elliptic curves and the classical and $p$-adic Birch and Swinnerton-Dyer conjectures by using these elements. The goal of this expository lecture note is to explain how Kato's Euler systems fit into the framework of the arithmetic of elliptic curves and their Iwasawa theory, and we hope that this approach eventually helps the reader to read Kato's original paper more easily and with less pain.
Comment: This is the refereed version. It is written for the proceedings of the ICTS program "Elliptic curves and the special values of $L$-functions". Several typos are corrected and the exposition is improved. Comments very welcome