Brain-inspired neuromorphic computing refers to simulation of the structure and functionality of the human brain via the integration of electronic or photonic devices. Artificial synapses are the most abundant computation element in the brain-inspired system. Memristors are considered to be ideal devices for artificial synapse applications because of their high scalability and low power consumption. Based on Ohm’s law and Kirchhoff’s law, memristor crossbar arrays can perform parallel multiply-accumulate operations in situ, leading to analogue computing with greatly improved speed and energy efficiency. Oxides are most widely used in memristors due to the ease of fabrication and high compatibility with CMOS processes. This work reviews the research progress of oxide memristors for brain-inspired computing, mainly focusing on their resistance switching mechanisms, device structures and performances. These devices fall into three categories: electrical memristors, memristors controlled via both electrical and optical stimuli, and all-optically controlled memristors. The working mechanisms of electrical memristors are commonly related to microstructure change and Joule heat that are detrimental to device stability. The device performance can be improved by optimizing device structure and material composition. Tuning the device conductance with optical signals can avoid microstructure change and Joule heat as well as reducing energy consumption, thus making it possible to address the stability problem. In addition, optically controlled memristors can directly response to external light stimulus enabling integrated sensing-computing-memoring within single devices, which are expected to be used for developing next-generation vision sensors. Hence, the realization of all-optically controlled memristors opens a new window for research and applications of memristors.