  This paper proves the Beilinson-Soul{\'e} vanishing conjecture for motives
attached to the moduli spaces of curves of genus 0 with n marked points. As
part of the proof, it is also proved that these motives are mixed Tate. As a
consequence of Levine's work, one obtains then well defined categories of mixed
Tate motives over the moduli spaces of curves . It is shown that morphisms
between moduli spaces forgetting marked points and embedding as boundary
components induce functors between those categories and how tangential bases
points fit in these functorialities. Tannakian formalism attaches groups to
these categories and morphisms reflecting the functorialities leading to the
definition of a motivic Grothendieck-Teichm{\"u}ller group. Proofs of the above
properties rely on the geometry of the tower of the moduli spaces . This allows
us to treat the general case of motives over Spec(Z) with integral coefficients
working in Spitzweck's category of motives. From there, passing to
Q-coefficients we deal with the classical tannakian formalism and explain how
working over Spec(Q) allows a more concrete description of the tannakian group.
