For the flexible job shop scheduling problem(FJSP) with due date, a discrete cat swarm optimization(DCSO) algorithm is proposed to optimize the makespan and the mean earliness/tardiness. Firstly, a two-phase discrete encoding approach is designed to represent the scheduling solution, and a heuristic-based method is adopted to fulfill the population initialization. Then, in order to make the algorithm work directly in a discrete scheduling space, a seeking method is developed based on three different neighborhood structures in the seeking mode, and a new discrete individual updating equation is proposed in the tracking mode. To coordinate the global and local search capability of the proposed algorithm, a linear adaptive selection strategy of cat's behavior modes is used. In addition, a local search strategy is embedded into the algorithm to further improve the computational results. Finally, the performance of the DCSO algorithm is tested based on the benchmark instances, and experimental data demonstrate the effectiveness of the proposed DCSO algorithm on solving the FJSP.