The so-called  ``sphere theorem'' asserts that a positively curved simply connected Riemannian manifold such that the ratio between the minimum and the maximum of the sectional curvature is greater than $1/4$ is homeomorphic to a sphere. Notice that  $1/4$ is the value of this ratio for the complex projectif spaces of complex dimension greater than one. This result was eventually proved in the sixties by M.~Berger and W.~Klingenberg, following a seminal article by H.E.~Rauch in 1951. It is only recently that S.~Brendle and R.~Schoen were able to prove that such a manifold is indeed diffeomorphic to a sphere; this is the differentiable sphere theorem that we shall describe in this note. The proof uses the Ricci flow, introduced by R.~Hamilton and an approach developped by Ch.~Böhm and B.~Wilking.