A new model-free approach for the description of general dynamicalsystems with unknownstructure, order, and excitation is introduced. The approach is based on the new concept of eigenvalue orbit. The eigenorbits are obtained by building an associated linear time-variant system through a matrix that relates the output measurements in a moving horizon window and viewing the trajectories of its time-varying eigenvalues. How the eigenorbits may be computed from the measurements and used for the characterization of the original system is shown. The basic properties of the eigenorbits are presented via a series of theorems for the case of a discrete-time, linear timeinvariant system. A set of examples are included to illustrate these properties for more general classes of systems and to suggest some practical issues that can be drawn from the orbits.