Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2017, Director: Francesc Xavier Massaneda Clares[en] Maps are a useful tool to display information. They are used on a daily basis to locate places, orient ourselves or present different features, such as weather forecasts, population distributions, etc. However, every map is a representation of the Earth that actually distorts reality. Depending on the purpose of the map, the interest may rely on preserving different features. For instance, it might be useful to design a map for navigation in which the directions represented on the map at a point concide with the ones the map reader observes at that point. Such map projections are called conformal. This dissertation aims to study different conformal representations of the Earth. The shape of the Earth is modelled by a regular surface. As both the Earth and the flat piece of paper onto which it is to be mapped are two-dimensional surfaces, the map projection may be described by the relation between their coordinate systems. For some mathematical models of the surface of the Earth it is possible to define a parametrization that verifies the conditions E = G and F = 0, where E, F and G denote the coefficients of the first fundamental form. In this cases, the mapping problem is shown to reduce to the study of conformal functions from the complex plane onto itself. In particular, the Schwarz-Christoffel formula for the mapping of the upper half-plane on a polygon is applied to cartography.