Networks of spiking neurons underpin the extraordinary information-processing capabilities of the brain and have emerged as pillar models in neuromorphic intelligence. Despite extensive research on spiking neural networks (SNNs), most are established on deterministic models. Integrating noise into SNNs leads to biophysically more realistic neural dynamics and may benefit model performance. This work presents the noisy spiking neural network (NSNN) and the noise-driven learning rule (NDL) by introducing a spiking neuron model incorporating noisy neuronal dynamics. Our approach shows how noise may act as a resource for computation and learning and theoretically provides a framework for general SNNs. Moreover, NDL provides an insightful rationale for surrogate gradients. By incorporating various SNN architectures and algorithms, we show that our approach exhibits competitive performance and improved robustness against challenging perturbations than deterministic SNNs. Additionally, we demonstrate the utility of the NSNN model for neural coding studies. Overall, NSNN offers a powerful, flexible, and easy-to-use tool for machine learning practitioners and computational neuroscience researchers.