Given a full subcategory[Fscr] of a category [Ascr], the existenceof left [Fscr]-approximations (or[Fscr]-preenvelopes) completing diagrams in a unique way isequivalent to the fact that [Fscr] is reflective in[Ascr], in the classical terminology of categorytheory.

In the first part of the paper we establish, for a rather general[Ascr], the relationship between reflectivity and covariantfiniteness of [Fscr] in [Ascr], andgeneralize Freyd's adjoint functor theorem (for inclusion functors) to notnecessarily complete categories. Also, we study the good behaviour of reflectionswith respect to direct limits. Most results in this part are dualizable, thusproviding corresponding versions for coreflective subcategories.

In thesecond half of the paper we give several examples of reflective subcategories ofabelian and module categories, mainly of subcategories of the form Copres(M) and Add (M). The second case covers thestudy of all covariantly finite, generalized Krull-Schmidt subcategories of{\rm Mod}_{R}, and has some connections with the“pure-semisimple conjecture”.

1991 Mathematics Subject Classification 18A40, 16D90, 16E70.