We study the size of the near-critical window for Bernoulli percolation on \mathbb Z^d. More precisely, we use a quantitative Grimmett-Marstrand theorem to prove that the correlation length, both below and above criticality, is bounded from above by \exp(C/|p-p_c|^2). Improving on this bound would be a further step towards the conjecture that there is no infinite cluster at criticality on \mathbb Z^d for every d\ge2.
