Given a map between compact Riemann surfaces, the Riemann-Hurwitz formula relates the Euler characteristics of the surfaces and the degree of the map. In this talk we will discuss the various components of this formula towards giving a sketch of the proof, and seeing some simple consequences in Topology and Algebraic Curves. We will define CW complexes in order to compute the Euler characteristic of surfaces, and then analyse the rigid structure of maps between Riemann surfaces to define the degrees of such maps, before seeing how to relate these numbers. I intend this to be a very accessible talk which will introduce some interesting ideas from Topology and Algebraic Curves. There will be lots of pictures!