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Inspired by the operation of biological brains, Spiking Neural Networks (SNNs) have the unique ability to detect information encoded in spatio-temporal patterns of spiking signals. Examples of data types requiring spatio-temporal processing include logs of time stamps, e.g., of tweets, and outputs of neural prostheses and neuromorphic sensors. In this paper, the second of a series of three review papers on SNNs, we first review models and training algorithms for the dominant approach that considers SNNs as a Recurrent Neural Network (RNN) and adapt learning rules based on backpropagation through time to the requirements of SNNs. In order to tackle the non-differentiability of the spiking mechanism, state-of-the-art solutions use surrogate gradients that approximate the threshold activation function with a differentiable function. Then, we describe an alternative approach that relies on probabilistic models for spiking neurons, allowing the derivation of local learning rules via stochastic estimates of the gradient. Finally, experiments are provided for neuromorphic data sets, yielding insights on accuracy and convergence under different SNN models.