Neural network–based models have shown tremendous achievements in various fields. However, standard AI‐based systems suffer from catastrophic forgetting when undertaking sequential learning of multiple tasks in dynamic environments. Continual learning has emerged as a promising approach to address catastrophic forgetting. It enables AI systems to learn, transfer, augment, fine‐tune, and reuse knowledge for future tasks. The techniques used to achieve continual learning are inspired by the learning processes of the human brain. In this study, we present a comprehensive review of research and recent developments in continual learning, highlighting key contributions and challenges. We discuss essential functions of the biological brain that are pivotal for achieving continual learning and map these functions to the recent machine‐learning methods to aid understanding. Additionally, we offer a critical review of five recent types of continual learning methods inspired by the biological brain. We also provide empirical results, analysis, challenges, and future directions. We hope that this study will benefit both general readers and the research community by offering a complete picture of the latest developments in this field.