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We prove an nÂ¿(-1)/4k lower bound on the randomized k-party communication complexity of depth 4 AC0 functions in the number-on-forehead (NOF) model for up to Â¿(log n) players. These are the first non-trivial lower bounds for general NOF multiparty communication complexity for any AC0 function for Â¿ (log log n) players. For non-constant k the bounds are larger than all previous lower bounds for any AC0 function even for simultaneous communication complexity. Our lower bounds imply the first superpolynomial lower bounds for the simulation of AC0 by MAJ o SYMM o AND circuits, showing that the well-known quasipolynomial simulations of AC0 by such circuits are qualitatively optimal, even for formulas of small constant depth. We also exhibit a depth 5 formula in NPk cc - BPPk cc for k up to Â¿(log n) and derive an Â¿(2Â¿(log n/ Â¿(k))) lower bound on the randomized k-party NOF communication complexity of set disjointness for up to Â¿(log1/3 n) players which is significantly larger than the O (log log n) players allowed in the best previous lower bounds for multiparty set disjointness. We prove other strong results for depth 3 and 4 AC0 functions.