The basic objects of study in algebraic number theory -- number fields, are classified by compact topological group called the \emph{absolute Galois group} of $\mathbf{Q}$, denoted $G_\mathbf{Q}$. Understanding the structure of $G_\mathbf{Q}$ is one of the main goals of number theory, but it remains very mysterious. In Grothendieck's letter "Esquisse d'un Programme", he outlined a new strategy for describing $G_\mathbf{Q}$ as a concrete subgroup of the outer automorphism group of the free group on two generators. Surprisingly, this approach uses ideas from topology and complex geometry. I will start by describing the formalism of algebraic fundamental groups with as little technicalities as possible. Then I will introduce the Teichmüller tower and say a bit about how $G_\mathbf{Q}$ embeds into the Grothendieck-Teichmüller group $\widehat{GT}$.