Reflective subcategories

Reflective subcategories Given a full subcategory 3 of a category A, the existence of left 3-approximations (or 3-preenvelopes) completing diagrams in a unique way is equivalent to the fact that 3 is reflective in A, in the classical terminology of category theory. In the first part of the paper we establish, for a rather general A, the relationship between reflectivity and covariant finiteness of 3 in A, and generalize F'reyd's adjoint functor theorem (for inclusion functors) to not necessarily complete categories. Also, we study the good behaviour of reflec-tions with respect to direct limits. Most results in this part are dualizable, thus providing corresponding versions for coreflective subcategories. In the second half of the paper we give several examples of reflective subcategories of abelian and module categories, mainly of subcategories of the form Copres (M) and Add (M). The second case covers the study of all covariantly finite, generalized Krull-Schmidt subcategories of ModR, and has some connections with the "pure-semisimple conjecture"