The notion of arrow structure /a.s./ is introduced as an algebraic version of the notion of directed multi graph. By means of a special kind of a representation theorem for arrow structures it is shown that the whole information of an a.s. is contained in the set of his arrows equipped with four binary relations describing the four possibilities for two arrows to have a common point. This makes possible to use arrow structures as a semantic base for a special polymodal logic, called in the paper BAL /Basic Arrow Logic/. BAL and various kinds of his extensions are used for expressing in a modal setting different properties of arrow structures. Several kinds of completeness theorems for BAL and some other arrow logics are proved, including completeness with respect to classes of finite models. And the end some open problems and possibilities for further development of the “arrow” approach are formulated.