n this article we study the whole class QSE of non-degenerate planar quadratic differential systems possessing at least one invariant ellipse.We classify this family of systems according to their geometric properties encoded in the configurations of invariant ellipses and invariant straight lines which these systems could possess. The classification, which staken modulo the action of t he group of real affine transformations and time rescaling, is given in terms of algebraic geometric invariants and also in terms of invariant polynomials and it yields a total of 35 distinct such configurations. This classification is also an algorithm which makes it possible to verify for any given real quadratic differential system if it has invariant ellipses or not and to specify its configuration of invariant ellipses and straight lines.