Positioning and stabilizing a catheter at the required location inside a vessel or the heart is a complicated task in interventional cardiology. In this review we provide a structured classification of catheter stabilization mechanisms to systematically assess their challenges during cardiac interventions. Commercially available, patented, and experimental prototypes of catheters were classified with respect to their stabilizing mechanisms. Subsequently, the classification was used to define requirements for future cardiac catheters and persisting challenges in catheter stabilization. The classification showed that there are two main stabilization mechanisms: surface-based and volume-based. Surface-based mechanisms apply attachment through surface anchoring, while volume-based mechanisms make use of locking through shape or force against the vessel or cardiac wall. The classification provides insight into existing catheter stabilization mechanisms and can possibly be used as a tool for future design of catheter stabilization mechanisms to keep the catheter at a specific location during an intervention. Additionally, insight into the requirements and challenges for catheter stabilization inside the heart and vasculature can lead to the development of more dedicated systems in the future, allowing for intervention- and patient-specific instrument manipulation.