We combine the concepts of modal logics and many-valued logics in a general and comprehensive way. Namely, given any finite linearly ordered set of truth values and any set of propositional connectives defined by truth tables, we define the many-valued minimal normal modal logic, presented as a Gentzen-like sequent calculus, and prove its soundness and strong completeness with respect to many-valued Kripke models. The logic treats necessitation and possibility independently, i.e., they are not defined by each other, so that the duality between them is reflected in the proof system itself. We also prove the finite model property (that implies strong decidability) of this logic and consider some of its extensions. Moreover, we show that there is exactly one way to define negation such that De Morgan's duality between necessitation and possibility holds. We also prove that many-valued intuitionistic logic is a fragment of one of the extensions of our many-valued modal logic.