The finite-time Mittag-Leffler synchronization is investigated for fractional-order delayed memristive neural networks (FDMNN) with parameters uncertainty and discontinuous activation functions. The relevant results are obtained under the framework of Filippov for such systems. Firstly, the novel feedback controller, which includes the discontinuous functions and time delays, is proposed to investigate such systems. Secondly, the conditions on finite-time Mittag-Leffler synchronization of FDMNN are established according to the properties of fractional-order calculus and inequality analysis technique. At the same time, the upper bound of the settling time for Mittag-Leffler synchronization is accurately estimated. In addition, by selecting the appropriate parameters of the designed controller and utilizing the comparison theorem for fractional-order systems, the global asymptotic synchronization is achieved as a corollary. Finally, a numerical example is given to indicate the correctness of the obtained conclusions.