Abstract: This talk is aimed at graduate students and is more or less an advertisement for the power of algebraic geometry. Two diverse theorems in completely different subjects will be unified in this talk. First we have the Galois correspondence in field theory. Second we have a topological theorem, the correspondence between covering spaces and subgroups of the fundamental group. Riemann's existence theorem, due to Grothendieck, is a kind of meta-theorem unifying these results. It leads to the notion of the etale fundamental group.