The Fontaine-Wintenberger theorem relates the absolute Galois group of $\mathbb{Q}_p(\mu_{p^{\infty}})$ to the absolute Galois group of a certain local field over $\mathbb{F}_p$. Such an equivalence enables us to understand the Galois representation of $\mathbb{Q}_p$ with some extra work. In this talk I will explain several key ideas behind its classical proof and its generalization via Faltings's almost purity theorem if time permits.