summary:Let G be an abstract class (closed under isomorpic copies) of left R-modules. In the first part of the paper some sufficient conditions under which G is a precover class are given. The next section studies the G-precovers which are G-covers. In the final part the results obtained are applied to the hereditary torsion theories on the category on left R-modules. Especially, several sufficient conditions for the existence of σ-torsionfree and σ-torsionfree σ-injective covers are presented