This paper studies what we call normal multimodal logics, which are general modal systems with an arbitrary set of normal modal operators. We emphasize the importance of non-simple systems, for which some interaction axioms are considered. A list of such acceptable axioms is proposed, among which the induction axiom has a special behavior. The class of multimodal logics that can be built with these axioms generalizes many existing modal, temporal, dynamic and epistemic systems, and could also suggest new formalizations using modal logics. The main result is a general determination theorem for these multimodal systems, which establishes a correspondence between our axioms and conditions over Kripke frames; this should avoid the need for showing determination each time a new system is considered.