Intuitionistic Modal Logic

0 Intuitionistic modal logics originate from di erent sources and have di erent areas of application. They include philosophy (see e.g. Prior [1957], Ewald [1986], Williamson [1992]), the foundations of mathematics (Kuznetsov [1985], Kuznetsov and Muravitskij [1986]), and computer science (Plotkin and Stirling [1986], Stirling [1987], Wijesekera [1990]). Modalities are added to intuitionistic logic in the framework of studying \new intuitionistic connectives" (Bessonov [1977], Gabbay [1977], Yashin [1994]) and to simulate the monadic fragment of intuitionistic rst order logic (Bull [1966], Ono [1977], Ono and Suzuki [1988], Bezhanishvili [1997]). The multitude of constructed logics was examined \piecewise", often by means of creating a special semantical and syntactical apparatus. A broader perspective is to try combining the well-developed general theories of classical modal logics and non-modal superintuitionistic (alias intermediate) logics in order to embrace the classes of extensions of some reasonable basic modal systems on the intuitionistic base. In this paper we give an overview of the model theory that results from such a combination and demonstrate a number of applications.