Networks of spiking neurons underpin the extraordinary information-processing capabilities of the brain and have become pillar models in neuromorphic artificial intelligence. Despite extensive research on spiking neural networks (SNNs), most studies are established on deterministic models, overlooking the inherent non-deterministic, noisy nature of neural computations. This study introduces the noisy SNN (NSNN) and the noise-driven learning (NDL) rule by incorporating noisy neuronal dynamics to exploit the computational advantages of noisy neural processing. The NSNN provides a theoretical framework that yields scalable, flexible, and reliable computation and learning. We demonstrate that this framework leads to spiking neural models with competitive performance, improved robustness against challenging perturbations compared with deterministic SNNs, and better reproducing probabilistic computation in neural coding. Generally, this study offers a powerful and easy-to-use tool for machine learning, neuromorphic intelligence practitioners, and computational neuroscience researchers.