The Riemann-Hurwitz formula
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In this thesis we discuss Riemann surfaces and (possibly ramified) coverings. We then discuss and prove the Riemann-Hurwitz formula. One of the applications of the Riemann-Hurwitz formula is that it can be used to prove Hurwitz's automorphisms theorem. We discuss this theorem and subsequently prove it. This thesis is ended by giving a short discussion on the equivalence of categories of compact Riemann surfaces and irreducible, non-singular algebraic curves over the complex plane.