Deep learning has generated diverse perspectives in astronomy, with ongoing discussions between proponents and skeptics motivating this review. We examine how neural networks complement classical statistics, extending our data analytical toolkit for modern surveys. Astronomy offers unique opportunities through encoding physical symmetries, conservation laws, and differential equations directly into architectures, creating models that generalize beyond training data. Yet challenges persist as unlabeled observations number in billions while confirmed examples with known properties remain scarce and expensive. This review demonstrates how deep learning incorporates domain knowledge through architectural design, with built-in assumptions guiding models toward physically meaningful solutions. We evaluate where these methods offer genuine advances versus claims requiring careful scrutiny. - Neural architectures overcome bias-variance trade-offs among scalability, expressivity, and data efficiency by encoding physical symmetries and conservation laws into network structure, enabling learning from limited labeled data. - Simulation-based inference and anomaly detection extract information from complex, non-Gaussian distributions where analytical likelihoods fail, enabling field-level cosmological analysis and systematic discovery of rare phenomena. - Multiscale neural modeling bridges resolution gaps in astronomical simulations, learning effective subgrid physics from expensive high-fidelity runs to enhance large-volume calculations where direct computation remains prohibitive. - Emerging paradigms-reinforcement learning for telescope operations, foundation models learning from minimal examples, and large language model agents for research automation-show promise though are still developing in astronomical applications.