  We review some ideas of Grothendieck and others on actions of the absolute
Galois group {\Gamma} Q of Q (the automorphism group of the tower of finite
extensions of Q), related to the geometry and topology of surfaces (mapping
class groups, Teichm{\"u}ller spaces and moduli spaces of Riemann surfaces).
Grothendieck's motivation came in part from his desire to understand the
absolute Galois group. But he was also interested in Thurston's work on
surfaces, and he expressed this in his Esquisse d'un programme, his
R{\'e}coltes et semailles and on other occasions. He introduced the notions of
dessin d'enfant, Teichm{\"u}ller tower, and other related objects, he
considered the actions of {\Gamma} Q on them or on their etale fundamental
groups, and he made conjectures on some natural homomorphisms between the
absolute Galois group and the automor-phism groups (or outer automorphism
groups) of these objects. We mention several ramifications of these ideas, due
to various authors. We also report on the works of Sullivan and others on
nonlinear actions of {\Gamma} Q , in particular in homotopy theory. The final
version of this paper will appear as a chapter in Volume VI of the Handbook of
Teichm{\"u}ller theory. This volume is dedicated to the memory of Alexander
Grothendieck.
