We will discuss a non-pointed version of the notion of torsion theory, in the framework of categories equipped with a posetal monocoreflective subcategory such that the coreflector inverts monomorphisms. We will explore the relationships of such torsion theories with factorization systems and categorical Galois structures. We will describe several examples of such torsion theories, in the duals of elementary toposes, in varieties of universal algebras used as models for non-classical logic, and in coslices of the category of abelian groups. Joint work with Andrea Cappelletti.