The Riemann zeta function is a fundamental topic in number theory. In this talk, our primary focus will be on the behavior of extreme values with the zeta function. We will first provide an introduction to the research history, covering various aspects as the behavior on the 1-line, within the critical strip, and on the critical line. We will also include results of the derivatives of the zeta function. Then we will present some recent results. The talk is mainly based on some joint work with Zikang Dong.