The present paper develops a theory of multistep natural continuous extensions of Runge-Kutta methods, that is interpolants of multistep type that generalize the notion of natural continuous extension introduced by Zennaro [15]. The main motivation for the definition of such a type of interpolants is given by the need for interpolation procedures with strong stability properties and high order of accuracy, in view of interesting applications to the numerical solution of delay differential equations and to the waveform relaxation methods for large systems of ordinary differential equations.
