Consider the class QS of all non-degenerate quadratic systems. Note that each quadratic polynomial differential system can be identiffed with a point of R12 through its coeffcients. In this paper we provide necessary and suffcient conditions for a system in QS, in term of its coeffcients, to have at least one invariant ellipse. Let QSE be the whole class of non-degenerate planar quadratic differential systems possessing at least one invariant ellipse. For the class QSE, we give the global \bifurcation" diagram which indicates where an ellipse is present or absent and in case it is present, the diagram indicates if the ellipse is or not a limit cycle. The diagram is expressed in terms of affne invariant polynomials and it is done in the 12-dimensional space of parameters. This diagram is also an algorithm for determining for each quadratic system if it possesses an invariant ellipse and whether or not this ellipse is a limit cycle.