Most current Artificial Neural Network (ANN) models are based on highly simplified brain dynamics. They have been used as powerful computational tools to solve complex pattern recognition, function estimation, and classification problems. ANNs have been evolving towards more powerful and more biologically realistic models. In the past decade, Spiking Neural Networks (SNNs) have been developed which comprise of spiking neurons. Information transfer in these neurons mimics the information transfer in biological neurons, i.e., via the precise timing of spikes or a sequence of spikes. To facilitate learning in such networks, new learning algorithms based on varying degrees of biological plausibility have also been developed recently. Addition of the temporal dimension for information encoding in SNNs yields new insight into the dynamics of the human brain and could result in compact representations of large neural networks. As such, SNNs have great potential for solving complicated time-dependent pattern recognition problems because of their inherent dynamic representation. This article presents a state-of-the-art review of the development of spiking neurons and SNNs, and provides insight into their evolution as the third generation neural networks.