Medical image analysis has witnessed remarkable advancements, even surpassing human-level performance in recent years, driven by the rapid development of advanced deep-learning algorithms. However, when the inference dataset slightly differs from what the model has seen during one-time training, the model performance is greatly compromised. The situation requires restarting the training process using both the old and the new data, which is computationally costly, does not align with the human learning process, and imposes storage constraints and privacy concerns. Alternatively, continual learning has emerged as a crucial approach for developing unified and sustainable deep models to deal with new classes, tasks, and the drifting nature of data in non-stationary environments for various application areas. Continual learning techniques enable models to adapt and accumulate knowledge over time, which is essential for maintaining performance on evolving datasets and novel tasks. Owing to its popularity and promising performance, it is an active and emerging research topic in the medical field and hence demands a survey and taxonomy to clarify the current research landscape of continual learning in medical image analysis. This systematic review paper provides a comprehensive overview of the state-of-the-art in continual learning techniques applied to medical image analysis. We present an extensive survey of existing research, covering topics including catastrophic forgetting, data drifts, stability, and plasticity requirements. Further, an in-depth discussion of key components of a continual learning framework, such as continual learning scenarios, techniques, evaluation schemes, and metrics, is provided. Continual learning techniques encompass various categories, including rehearsal, regularization, architectural, and hybrid strategies. We assess the popularity and applicability of continual learning categories in various medical sub-fields like radiology and histopathology. Our exploration considers unique challenges in the medical domain, including costly data annotation, temporal drift, and the crucial need for benchmarking datasets to ensure consistent model evaluation. The paper also addresses current challenges and looks ahead to potential future research directions.