In his proof of Apery’s theorem on the irrationality of $\zeta \left(3\right)$, Beukers [B] introduced double and triple integrals of suitable rational functions yielding good sequences of rational approximations to $\zeta \left(2\right)$ and $\zeta \left(3\right)$. Beukers’ method was subsequently improved by Dvornicich and Viola, by Hata, and by Rhin and Viola. We give here a survey of our recent results ([RV2] and [RV3]) on the irrationality measures of $\zeta \left(2\right)$ and $\zeta \left(3\right)$ based upon a new algebraic method involving birational transformations and permutation groups...