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As semiconductor technology enters the more than Moore era, there exists an apparent contradiction between the rapidly growing demands for data processing and the visible inefficiency rooted in traditional computing architecture. Neuromorphic systems hold great prospects in enabling a new generation of computing paradigm that can address this issue, which demands device components with rich dynamics and nonlinearity. Herein, the nonlinearity in memristive devices and their application in building neuromorphic dynamic systems are reviewed. The internal mechanisms that endow memristive devices with nonlinearity and rich dynamics are reviewed and subsequently the nonlinear spiking neurons that are implemented utilizing the physical processes in memristors are shown. Typical examples on neuromorphic dynamic systems based on nonlinear memristors are summarized, including memristive reservoir, memristive oscillatory neural network, and memristive chaotic computing. Finally, an outlook in the development of neuromorphic dynamic systems is given.