After giving a brief introduction to Q-algebra of real multizeta values, we will recall the main features of the geometry of the genus zero moduli spaces of curves and some Grothendieck-Teichmuller theory in genus zero, and show how multizeta values tie in with this. Indeed, the Drinfeld associator, a power series whose coefficients are all multizeta values, is well-known to satisfy three fundamental relations analogous to those of the Grothendieck-Teichmuller group. But in fact, the MZV's arise as periods of the genus zero moduli spaces, which sheds new light on this fact. It also gives rise to some very natural new conjectures, for instance: "all periods of the genus zero moduli spaces satisfy analogous relations", or more strongly, "all periods of the genus zero moduli spaces are linear combinations of multizeta values" (Goncharov-Manin conjecture). If there is time, we will give a brief introduction to mixed Tate motives and explain how multizeta values arise in this context, via the cohomology of the genus zero moduli spaces, and explain Goncharov's fundamental conjectures.