We review our recent results on the emergence of extreme events in networks of model neurons. We first focus on a collection of aperiodically spiking neurons, under local diffusive coupling, as well as global coupling through the collective mean field. Our principal finding is that the occurrences of both temporal and spatial extreme events are augmented by local diffusive coupling. In pronounced contrast, extreme events are annihilated by mean-field coupling. This indicates the profound impact of the nature of coupling on extreme spiking events in coupled neurons. Further, we review the effect of non-local diffusive coupling on extreme events. Here interestingly, we find a large window where spatial synchronization exists alongside temporal extreme events, i.e. we obtain synchronized extreme spiking, reminiscent of the neuronal activity during epileptic seizures. Finally, we review results on the influence of random links on the advent of extreme events. We demonstrate that when coupling is weak, random links suppress extreme events. However, when coupling is strong and there exist a sufficient number of static random links, the system is prone to extreme events. We also find that when the links vary in time, even a small number of random links can induce extreme events in a large window of coupling. Examining the effect of link-rewiring frequency reveals that, even infrequently switched links can generate significantly more extreme events than static networks. These results have broad bearing on extreme events in time-varying complex systems in general, as well as specific implications for time-varying networks of model neurons. Lastly, importantly, we find that the variability in the probability of extreme events gets more pronounced at the verge of cross-overs to regimes yielding extreme events. So, this feature can be utilized as an early warning signal for extreme events, and potentially aid risk appraisal.