Recent advancements in artificial intelligence (AI), particularly foundation models (FMs), have revolutionized medical image analysis, demonstrating strong zero- and few-shot performance across diverse medical imaging tasks, from segmentation to report generation. Unlike traditional task-specific AI models, FMs leverage large corpora of labeled and unlabeled multimodal datasets to learn generalized representations that can be adapted to various downstream clinical applications with minimal fine-tuning. However, despite the rapid proliferation of FM research in medical imaging, the field remains fragmented, lacking a unified synthesis that systematically maps the evolution of architectures, training paradigms, and clinical applications across modalities. To address this gap, this review article provides a comprehensive and structured analysis of FMs in medical image analysis. We systematically categorize studies into vision-only and vision-language FMs based on their architectural foundations, training strategies, and downstream clinical tasks. Additionally, a quantitative meta-analysis of the studies was conducted to characterize temporal trends in dataset utilization and application domains. We also critically discuss persistent challenges, including domain adaptation, efficient fine-tuning, computational constraints, and interpretability along with emerging solutions such as federated learning, knowledge distillation, and advanced prompting. Finally, we identify key future research directions aimed at enhancing the robustness, explainability, and clinical integration of FMs, thereby accelerating their translation into real-world medical practice.