Although “complexity” in health care is commonly associated with “tertiary” hospitals in many countries, there is no accepted defini- tion of tertiariness or agreement on structural and funding arrange- ments. This paper attempts a conceptual analysis of complexity in tertiary hospitals in terms of clinical views, capability requirements and process-orientations. The main insight is that we need a de- mand- and process based concept of tertiariness because a supply- side concept is not operational. The paper also develops some ideas what such a demand-based concept would look like.