Neuromorphic computing is one of the most promising technologies to solve the von Neumann bottleneck, which has the advantages of fast processing speed and low energy consumption in performing complex tasks. The development of neuromorphic computing is currently driven by several kinds of novel devices. Magnetic tunnel junctions (MTJs) are rich in nonlinear properties and can be regulated by multiple physical fields such as magnetic field, current and temperature. Meanwhile, MTJ has the advantages of good stability and low power consumption, which makes it an ideal device for neuromorphic computing. This paper starts by examining individual MTJ devices and then extends the discussion to full neural networks. First of all, we sorted out the various properties of MTJ, from the structure to physical mechanism and response characteristics. Secondly, the biological neuron model, synaptic properties and related studies on simulating neurons and synapses based on MTJs are introduced. Then, we review the neural network system-level architectures that have been explored with MTJ devices. Finally, the challenges and the future development trend are summarized for advancing MTJ-enabled neuromorphic computing.