Neural networks, one of the key artificial intelligence technologies today, have the computational power and learning ability similar to the brain. However, implementation of neural networks based on the CMOS von Neumann computing systems suffers from the communication bottleneck restricted by the bus bandwidth and memory wall resulting from CMOS downscaling. Consequently, applications based on large-scale neural networks are energy/ area hungry and neuromorphic computing systems are proposed for efficient implementation of neural networks. Neuromorphic computing system consists of the synaptic device, neuronal circuit, and neuromorphic architecture. With the two-terminal nonvolatile nanoscale memristor as the synaptic device and crossbar as parallel architecture, memristor crossbars are proposed as a promising candidate for neuromorphic computing. Herein, neuromorphic computing systems with memristor crossbars are reviewed. The feasibility and applicability of memristor crossbars based neuromorphic computing for the implementation of artificial neural networks and spiking neural networks are discussed and the prospects and challenges are also described.