We consider the development of a general nonlinear input-output theory which encompasses systems with initial conditions. Systems are defined in a set theoretic manner from input-output pairs on a doubly infinite time axis, and a general construction of the initial conditions is given in terms of an equivalence class of trajectories on the negative time axis. Input-output operators are then defined for given initial conditions, and a suitable notion of input-output stability on the positive time axis with initial conditions is given. This notion of stability is closely related to the ISS/IOS concepts of Sontag. A fundamental robust stability theorem is derived which represents a generalization of the input-output operator robust stability theorem of Georgiou and Smith, to include the case of initial conditions. This includes a suitable generalization of the nonlinear gap metric. Some applications are given to show the utility of the robust stability theorem.