Brain-inspired computing is an emerging field that aims at building a compact and massively parallel architecture, to reduce power consumption in conventional Von Neumann Architecture. Recently, memristive devices have gained great attention due to their immense potential in implementing brain-inspired computing and perception. The conductance of a memristor can be modulated by a voltage pulse, enabling emulations of both essential synaptic and neuronal functions, which are considered as the important building blocks for artificial neural networks. As a result, it is critical to review recent developments of memristive devices in terms of neuromorphic computing and perception applications, waiting for new thoughts and breakthroughs. The device structures, operation mechanisms, and materials are introduced sequentially in this review; additionally, late advances in emergent neuromorphic computing and perception based on memristive devices are summed up. Finally, the challenges that memristive devices toward high-performance brain-inspired computing and perception are also briefly discussed. We believe that the advances and challenges will lead to significant advancements in artificial neural networks and intelligent humanoid robots.