Coherence-spaces and domains with totality are used to give interpretations of inductively defined types. A category of coherence spaces with totality is defined and the closure of positive inductive type constructors is analysed within this category. Type streams are introduced as a generalisation of types defined by strictly positive inductive definition. A semantical analysis of type streams with continuous recursion theorems is established. A hierarchy of domains with totality defined by positive induction is defined, and density for a sub-hierarchy is (...) proved. (shrink)