What is normal?

On the basis of the definition of a normal modal system, it is common to distinguish between normal and non-normal modal operators. However, we argue that the standard definition of a normal operator in the literature leads to at least two problems. First, it induces a mismatch between the normality of a system and the attribution of normality to an operator of this system. Second, the standard definition is insensitive to different occurrences of the possibility operator in various normal and non-normal systems. We introduce a definition of normal operator which is not affected by either of these problems, and discuss its applications both to the monomodal and to the multimodal settings.