A smart aquaculture management system (SAMS) based on the Internet of Things (IoT) has recently gained much attention for fulfilling the growing demand for aquaculture products. The SAMS uses cutting-edge sensing technologies with a modern networking system to continuously monitor water quality, animal health, and feeding behavior to improve the production yield. This review aims to comprehensively summarize and discussed the recent advancements in IoT-based technologies, e.g., sensors nodes, software interfacing, artificial intelligence, cloud computing, and storage, to develop reliable and robust SAMS for small- and large-scale aquaculture production modules. A future perspective section is included at the end of this review to recommend the needed research to be done to scale up the related digital technologies for application in aquaculture farms. The knowledge in this thorough review can be useful for the scientific community for needful advancements and modifications of the IoT technologies and for existing and future aquaculture farmers to adopt modern SAMS for improving the benefit-cost ratios.