Spiking neural network is the most biologically plausible neural network model, which is the core component of brain-inspired intelligence. Firstly, the most commonly used spiking neuron models and the structures of feedforward and recurrent spiking neural networks are introduced, and the temporal coding method of spiking neural networks is introduced. Then, the learning algorithms of spiking neural networks systematically, including the unsupervised learning and supervised learning algorithms are introduced. The supervised learning algorithm follows the gradient descent algorithm, the algorithm based on STDP rules and the algorithm based on spike trains convolution rules are introduced and summarized in details. Subsequently, this survey gives application examples of spiking neural networks in the fields of control, pattern recognition and brain-inspired intelligence. Some case studies of spiking neural networks and neuromorphic processor in various national brain initatives are presented. Finally, this survey discusses the current difficulties and challenges of spiking neural networks.