An intelligent system must incrementally acquire, update, accumulate, and exploit knowledge to navigate the real world's intricacies. This trait is frequently referred to as Continual Learning (CL), and it can be limited by catastrophic forgetting, a phenomenon in which learning anew task acutely reduces the system's performance on prior tasks. Numerous strategies have been developed to address this issue, as CL is essential for developing Artificial Intelligence (AI) systems that adapt to dynamic environments. This study examines the practical applications of CL, concentrating on energy management systems and their integration with Deep Learning (DL) models. Energy management systems are strategies and methods for monitoring, controlling, and optimizing energy use within a system or organization. The literature is systematically analyzed, highlighting methods such as replay techniques, regularization strategies, and architectural adaptations that address the challenges of catastrophic forgetting. Moreover, the review encompasses various energy-related applications, including non intrusive load monitoring, demand-side management, fault/anomaly detection, load forecasting/prediction, and renewable energy integration. Additionally, a case study on anomaly detection in energy systems is conducted, comparing different CL approaches. The case study findings aim to bridge the gap between theoretical advancements and real-world applications, providing insights and guidelines for implementing CL in diverse fields. Finally, this survey identifies key challenges that impede the deployment of CL and suggests potential directions to enhance its implementation in the energy management sector.