This paper aims to investigate the finite-time and fixed-time robust synchronization of fuzzy Cohen-Grossberg neural networks with discontinuous activations. To deal with the discontinuous property, the framework of Filippov solution is invoked to solve the inexistence of the classical solutions. For the purpose of achieving fixed-time synchronization, we turn to investigating the fixed-time stability problem of the error system between the drive-response systems. By the functional differential inclusions theory, inequality technique and the non-smooth Lyapunov-Krasovskii functional and designing a simple discontinuous state-feedback control law for the response neural system, some sufficient algebraic criteria are derived to achieve synchronization within a fixed time, and the settling time is given. Moreover, based on the fixed-time robust synchronization and under the same basic assumptions, we further complete the finite-time synchronization of the addressed drive-response systems by designing a simple switching adaptive controller, and the upper bound of the settling time is also estimated. It should be regarded as the first time to study the finite-time synchronization of fuzzy Cohen-Grossberg neural networks with discontinuous activations based on the adaptive control. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed synchronization strategies.