Galois theory is the study of certain extensions of a base object by inspecting the symmetries they enjoy. Not too suprisingly, this vague idea can be applied in many areas of mathematics. In this seminar we will look at how this phenomenon arises in algebra, looking at extensions of fields, in topology, with topological coverings, and finally in differential geometry, with ramified extensions of Riemann surfaces. This will allow us to observe the similarities between the various contexts. Finally, we will use the extra differential structure of Riemann surfaces to translate some algebraic questions in topological terms, and see how this can be applied to carry out certain computations.