The Riemann-Roch theorem, proved in the middle of the nineteenth century, is one of the early highlights of complex geometry. The goal of the talk is to explain its statement, the somewhat enigmatic Riemann-Roch formula, and to show, by giving examples and applications, why it is so central to the theory of Riemann surfaces and to algebraic geometry. We will begin by explaining what a Riemann surface (a.k.a. complex manifold of dimension 1) is and how Riemann came up with the idea. We then move on to see what kinds of functions can naturally be found on a Riemann surface. The Riemann-Roch formula makes a statement about the number of linearly independent functions on a (compact) Riemann surface with certain properties. While this sounds somewhat dry, the implications of the formula are quite exciting; we will see several applications to the classification of Riemann surfaces and will also use the theorem to relate Riemann surfaces to algebraic geometry. I will only assume some basic complex analysis and will try to explain everything else.