Urban flooding constitutes the most pervasive and costly natural hazard confronting cities globally, with climate change intensifying precipitation extremes and overwhelming conventional drainage infrastructure designed for historical rainfall patterns. This review examines the transformation of urban stormwater management through climate-adaptive drainage systems integrating green–gray hybrid infrastructure and smart technologies. The objective is to synthesize current knowledge on flood-resilient infrastructure planning, encompassing sustainable drainage systems (SuDS), low-impact development (LID), and digitally enabled stormwater networks. Key frameworks assessed include green–gray integration models achieving peak flow reductions of 20–60% and runoff volume decreases of 30–90% depending on intervention type and spatial configuration. Smart technologies—IoT sensors, AI-based flood forecasting, and digital twins—enable real-time monitoring with prediction lead times of 30–120 minutes and automated control optimizing system capacity during extreme events. Policy applications span retrofitting existing urban fabric, mega-city infrastructure planning, and coastal adaptation, with benefit–cost ratios ranging from 1.5 to 5.6 for green infrastructure investments. The review identifies critical implementation challenges including institutional fragmentation, data interoperability, and environmental justice concerns, concluding that climate-adaptive drainage requires paradigm shift from hazard elimination to resilience-based management integrating technical innovation with adaptive governance.