Serre’s conjecture and its consequences

After some generalities about the absolute Galois group of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb Q$$\end{document}, we present the historical context in which Serre made his modularity conjecture. This was recently proved by Wintenberger and the author ([22], [23]), with an input of Kisin ([24]). The focus of these notes is on the applications of the conjecture. Some of the applications are based on the methods used in the proof.