ABSTRACT: We developed a general theory of phantom morphisms in additive exact categories associated to subfunctors of the Ext functor. In particular, when the exact category has enough projective and injective modules, we prove that an ideal I in the category is special precovering if and only if there exists a subfunctor F of Ext with enough injective morphisms such that I is the ideal of phantom morphisms associated to the subfunctor F. We will specially stress the connection of our results with the classical notion of approximations of modules due to Auslander and Enochs. We will also emphasize difffererent applications of this theory as, for instance, the existence of phantom covers of modules, the Auslander-Reiten theory for Artin Algebras, the homotopy theory of semisplit complexes of modules, or the study of geometrical purity in the category of quasi-coherent sheaves over a scheme.