Neuromorphic data carries information in spatio-temporal patterns encoded by spikes. Accordingly, a central problem in neuromorphic computing is training spiking neural networks (SNNs) to reproduce spatio-temporal spiking patterns in response to given spiking stimuli. Most existing approaches model the input-output behavior of an SNN in a deterministic fashion by assigning each input to a specific desired output spiking sequence. In contrast, in order to fully leverage the time-encoding capacity of spikes, this work proposes to train SNNs so as to match distributions of spiking signals rather than individual spiking signals. To this end, the paper introduces a novel hybrid architecture comprising a conditional generator, implemented via an SNN, and a discriminator, implemented by a conventional artificial neural network (ANN). The role of the ANN is to provide feedback during training to the SNN within an adversarial iterative learning strategy that follows the principle of generative adversarial network (GANs). In order to better capture multi-modal spatio-temporal distribution, the proposed approach -- termed SpikeGAN -- is further extended to support Bayesian learning of the generator's weight. Finally, settings with time-varying statistics are addressed by proposing an online meta-learning variant of SpikeGAN. Experiments bring insights into the merits of the proposed approach as compared to existing solutions based on (static) belief networks and maximum likelihood (or empirical risk minimization).