This will be mainly an expository talk aiming to explain the correspondence introduced by Grothendieck in the 80's between algebraic curves defined over number fields and a certain kind of graphs embedded in compact orientable surfaces that he called designs d'infants (= children's drawings).

Towards the end of the talk I will attempt to mention two recent results obtained jointly with Andrei Jaikin-Zapirain concerning the action of the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ on regular dessins and on the topology of algebraic varieties defined over number fields. Speaker(s): Gabino Gonzalez-Diez (Universidad Autonoma de Madrid)