The rapid rise of transformer-based large language models (LLMs) has introduced new opportunities for automation and decision support in radiology, particularly in applications such as report generation, protocol optimization, and structured interpretation. Despite their impressive performance in producing contextually coherent text, conventional LLMs remain limited by their inability to interact autonomously with external systems, retrieve data, or execute code, restricting their role in real-world clinical and research workflows. To address these limitations, agentic systems have emerged as a new paradigm. By embedding LLMs within frameworks that enable reasoning, planning, and action, agentic systems extend LLM capabilities to dynamic interaction with users, tools, and data sources. This review provides a comprehensive overview of the foundations, architectures, and operational mechanisms of agentic systems, focusing on their applications in medical imaging and radiology. It summarizes key developments in the literature, including recent multi-agent frameworks for automated radiomics pipelines, and discusses the potential benefits of these systems in enhancing the reproducibility, interpretability, and accessibility of AI-driven workflows. The review critically examines current regulatory considerations, ethical implications, and sustainability challenges to highlight essential gaps that must be addressed for the safe and responsible clinical integration of these systems.