In recent years, digital shapes have become more and more widespread and have been made available in a plethora of online repositories. A systematic and formal approach for capturing and representing shape-related information is needed to facilitate its reuse and enable the demonstration of useful cross-domain usage scenarios. In this paper we present an ontology for digital shapes, called the Common Shape Ontology (CSO). We discuss the rationale, the requirements and the scope of this ontology, we present in detail its structure and describe the most relevant choices related to its development. Finally, we show how the CSO conceptualization is used in domain-specific application scenarios.