Abstract. A folklore conjecture states that the values of the Riemann zeta function at odd positive integers are algebraically independent over $\Q$. The main results in this direction are that $\zeta(3)$ is irrational (Ap\'ery), and that there are infinitely many irrational numbers amongst $\zeta(2n+1)$ (Rivoal). It is not known, however, whether $\zeta(5)$ is irrational or not. In this talk, I will begin by surveying the known and conjectured diophantine properties of multiple zeta values. I will then explain Beukers' elementary proof that $\zeta(3)$ is irrational, and describe the group structures corresponding to $\zeta(2)$ and $\zeta(3)$, which give the best bounds for their irrationality measures known to date. In the last part of the talk I will explain how the moduli spaces of curves of genus 0 give a unifying geometric interpretation of these proofs.