After defining the curves complexes and recalling part of their geometric meaning, I will briefly review the existing results in the discrete setting and their relevance for the study of automorphisms of Teichmueller groups (more generally morphisms between their finite index subgroups). I will then move to the profinite setting and state the main results (possibly with very brief sketches of proofs) underlining on the way the parallels and contrasts with the disrete case. I will also explain how the Grothendieck-Teichmueller group arises in this setting.