We establish relations between two extensions of Dickson's torsion theory, one introduced by Barr and the other by Cassidy, Hubert and Kelly. These two notions are studied in a general category and we show that categories with zero object are, in some sense, the ones where the second notion is relevant. In these categories, under suitable completeness conditions, we characterize torsion theories in terms of reflections and connections. A characterization of the torsion-free subcategories which generalizes the one of Dickson is also presented.