Inspired by a question of Riecke, we consider the interaction of totality and full abstraction, asking whether full abstraction holds for Scott’s model of cpos and continuous functions if one restricts to total programs and total observations. The answer is negative, as there are distinct operational and denotational notions of totality. However, when two terms are each total in both senses then they are totally equivalent operationally i they are totally equivalent in the Scott model. Analysing further, we consider sequential and parallel versions of PCF and several models: Scott’s model of continuous functions, Milner’s fully abstract model of PCF and their e ective submodels. We investigate how totality di ers between these models. Some apparently rather di cult open problems arise, essentially concerning whether the sequential and parallel versions of PCF have the same expressive power, in the sense of total equivalence.