  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. It is written for the proceedings of the
program "Elliptic curves and the special values of $L$-functions (ONLINE)" in
ICTS in August 2021.
