The present text is an expanded version of the lecture notes for a course on Riemann surfaces and dessins d'enfants which the authors have taught for several years to students of the masters degree in mathematics at the Universidad Autónoma de Madrid.

Riemann surfaces are an ideal meeting ground for several branches of mathematics. For example, a student taking a course like this will encounter concepts of algebraic topology (fundamental group, theory of covering spaces, monodromy), elements of Riemannian geometry (geodesics, isometries, tessellations), objects belonging to algebra and algebraic geometry (field extensions, algebraic curves, valuations), definitions belonging to arithmetic geometry (fields of moduli and definition of an algebraic variety), some elementary graph theory (dessins d'enfants), tools of (complex) analysis (Weierstrass functions and Poincaré series) and some of the most relevant groups in analytic number theory (principal congruence subgroups).

One of the main features of the theory of Riemann surfaces is that there is a bijective correspondence between isomorphism classes of compact Riemann surfaces and isomorphism classes of complex algebraic curves. Establishing this correspondence requires proving first that a Riemann surface has enough meromorphic functions to separate its points. This can be done by either applying the Riemann–Roch Theorem or using the Uniformization Theorem to construct these functions by means of Poincaré series (Weierstrass functions, in the genus one case). In this book we have chosen the second option, thereby introducing Fuchsian groups, the third member of this trinity of equivalent objects.