Abstract. We show that several discrepancy-like problems can be solved in NC nearly achieving the discrepancies guaranteed by a probabilistic analysis and achievable sequentially. For example, we describe an NC algorithm that given a set system (X, S) , where X is a ground set and S2X , computes a set RX so that for each SS the discrepancy ||RS|-|R-S|| is [MATHEMATICAL FORMULA] . Whereas previous NC algorithms could only achieve discrepancies [MATHEMATICAL FORMULA] with 0 , ours matches the probabilistic bound within a multiplicative factor 1+o(1) . Other problems whose NC solution we improve are lattice approximation, -approximations of range spaces with constant VC-exponent, sampling in geometric configuration spaces, approximation of integer linear programs, and edge coloring of graphs.