Abstract: Subdivision surfaces allow smooth free-form surface modelling without topological constraints. They have become a fundamental representation for smooth geometry, particularly in the animation and entertainment industries. This survey summarizes research on subdivision surfaces over the last 15 years in three major strands: analysis, integration into existing systems and the development of new schemes. We also examine the reason for the low adoption of new schemes with theoretical advantages, explain why Catmull-Clark surfaces have become a de facto standard in geometric modelling, and conclude by identifying directions for future research.