From this point on, main issues in system theory are tackled. The very first, considered in this chapter, is the all-important question of system identification. This is perhaps the most basic question in system theory and related linear algebra, with a large pedigree starting from Kronecker's characterization of rational functions to its elegant solution for time-variant systems presented here. Identification, often also called realization, is the problem of deriving the internal system’s equations (called state-space equations) from input–output data. In this chapter, we only consider the causal, or block-lower triangular case, although the theory applies just as well to an anti-causal system, for which one lets the time run backward, applying the same theory in a dual form.