We study triangulated categories and torsion theories in them, and compare two definitions of torsion theories in this work. The most important types of torsion theories—weight structures and t-structures (and admissible triangulated subcategories) are also been considered. One of the aims of this paper is to show that a number of basic definitions and properties of weight and t-structures naturally extend to arbitrary torsion theories (in particular, we define smashing and cosmashing torsion theories). This can optimize some of the proofs. Similarly, the definitions of orthogonal and adjacent weight and t-structures are generalized. We relate the adjacency of torsion theories with Brown-Comenetz duality and Serre functors. These results may be applied to the study of t-structures in compactly generated triangulated categories and in derived categories of coherent sheaves. The relationship between torsion theories and projective classes is described.