Research in different areas over the past ten years has revealed surprising relations between certain Lie algebras arising in very different ways from different areas of number theory: the "Galois" Lie algebra coming from the action of the absolute Galois group over Q on the fundamental group of P^1-{0,1,infinity}, the Lie algebra associated to multiple zeta values and its formal version (known as double shuffle), the Lie algebra associated to the Grothendieck Teichmuller group, and the free Lie algebra which arises naturally as the Tannakian fundamental group of the category of mixed Tate motives over Z. In the present talk we review the conjectures and partial results linking these Lie algebras, underlining in particular the close relations of multiple zeta values with Galois theory.