Generative Artificial Intelligence (GAI) has rapidly emerged as a transformative force in aquaculture, enabling intelligent synthesis of multimodal data, including text, images, audio, and simulation outputs for smarter, more adaptive decision-making. As the aquaculture industry shifts toward data-driven, automation and digital integration operations under the Aquaculture 4.0 paradigm, GAI models offer novel opportunities across environmental monitoring, robotics, disease diagnostics, infrastructure planning, reporting, and market analysis. This review presents the first comprehensive synthesis of GAI applications in aquaculture, encompassing foundational architectures (e.g., diffusion models, transformers, and retrieval augmented generation), experimental systems, pilot deployments, and real-world use cases. We highlight GAI's growing role in enabling underwater perception, digital twin modeling, and autonomous planning for remotely operated vehicle (ROV) missions. We also provide an updated application taxonomy that spans sensing, control, optimization, communication, and regulatory compliance. Beyond technical capabilities, we analyze key limitations, including limited data availability, real-time performance constraints, trust and explainability, environmental costs, and regulatory uncertainty. This review positions GAI not merely as a tool but as a critical enabler of smart, resilient, and environmentally aligned aquaculture systems.