The extreme value theory for Gaussian logarithmically correlated fields has emerged in the last decade as a powerful tool in the analysis of interface models, quantum gravity, random matrices and in a myriad of other applications. The two dimensional Gaussian free field (and its discrete analogue) is an important motivating example of such a field. In this lecture, I will describe the relation and differences between the extreme value theory for i.i.d. variables and that for G.-LCFs, and discuss a sample of non-Gaussian examples. Links to PDEs, random walks, random matrices and the Riemann zeta function will be highlighted.