Spiking neural networks (SNNs) are inspired from natural computing, modelling with high accuracy the interactions and processes between the synapses of the neurons focusing on low response time and energy efficiency. This novel paradigm of event-based processing opens new opportunities for discovering applications and developing efficient learning methods that should highlight the advantages of SNNs such as the large memory capacity and the fast adaptation, while preserving the easy-to-use and portability of the conventional computing architectures. In this paper, we do a brief review of the developments of the past decades in the field of SNNs. We start with a brief history of the SNN and summarize the most common models of spiking neurons and methods to implement synaptic plasticity. We also classify the SNNs according to the implemented learning rules and network topology. We present the computational advantages, liabilities, and applications suitable for using SNNs in terms of energy efficiency and response time. In addition, we briefly sweep through the existing platforms and simulation frameworks for SNNs exploration. The paper ends with conclusions that show predictions of future challenges and the emerging research topics associated with SNNs.