We prove that the set disjointness problem has randomized communication complexity (n2kk) in the number-on-the-forehead model with k parties, a quadratic improvement on the previous bound (n2k)12 . Our result remains valid for quantum protocols, where it is essentially tight. Proving it was an open problem since 1997, even in restricted settings such as one-way classical protocols with k=4 parties. We obtain other near-optimal results for multiparty set disjointness, including an XOR lemma, a direct product theorem, and lower bounds for nondeterministic and Merlin-Arthurprotocols. The proof contributes a novel technique for lower bounds on multiparty communication, based on directional derivatives of communication protocols over the reals.