We consider the following question, known as the inverse Galois problem (IGP): Given a finite group $G$, is there a finite Galois extension of $\mathbb{Q}$ with Galois group isomorphic to $G$? For example, twenty-five out of the twenty-six sporadic groups have been realized as Galois groups over the rationals, with the famous exception of $M_{23}$. While this question would seem to reside only in algebra/number theory, techniques from topology, algebraic geometry, and complex analysis have proven to be effective in handling certain cases. In this talk we introduce the IGP and do some concrete examples, while examining how the connection between Galois groups and fundamental groups plays a role in the background. On our path we will run into what is known as Belyi's theorem, a result that Grothendieck found "deep and disconcerting."