I will introduce several typical computing models of neuron and synapses that can be utilized to build spiking neural networks (SNNs). Additionally, selected inference and learning algorithms for SNNs will be discussed, followed by a brief overview of existing hardware and software solutions for implementing neuromorphic computing. I will further present our Error-Modulated Spike-Timing-Dependent Plasticity (EMSTDP) algorithm and its implementation on a neurosynaptic processor.