An integral equation description for linear systems is developed and used as the basis for the development of various system identification, model reduction and order determination methods. The system integral equation is utilized in the problem of parameter identification in continuous linear single-input single-output, multi-input multi-output and linear in parameters nonlinear systems. The approach is developed in the time domain where the effect of non-zero initial conditions and additive disturbances occurs naturally. Parameter estimates are deduced using several weighted residual concepts which have previously been used to produce approximate solutions to differential equations.