Belyi maps are covers of the projective line ramified only above 0, 1, and the point at infinity. But that is merely their algebraic-geometric guise. They can be interpreted in many alternative and surprising ways, be it combinatorially, as triples of finite permutations , , such that , or topologically, as drawings on compact orientable surfaces. In his Esquisse d’un Programme, Grothendieck had high hopes to use these correspondences to better understand the absolute Galois group of .

In this talk, we describe the aforementioned correspondences in some detail. After this, we discuss how Belyi maps can be computed explicitly by using the theory of modular forms. Recently, the implementation of these techniques in a collaboration with its creators Musty, Schiavone and Voight has led to a database that contains 616 Belyi maps of degree up to 9. This database is freely available online in the L-Functions and Modular Forms Database.