A basic mathematical formalism of dynamical systems is presented from the view-point of control engineering. The notion of input-output systems is given in a general logistic format in which most of the detailed system models can be embedded. Concepts of time, time system, causality, and state are introduced. The time system is of a generalized form in which the time domains of component functions of input and output may be different from each other. The causality is a term for the traditional non-anticipation. The state is introduced firstly in a pure form. Then a state representation, state mapping, state transition, and a static output system are formalized. The results of discussion and propositions show close interrelations between the causality, state mapping, and static output system.