The paper is dedicated to triangulated categories along with torsion theories in them; we compare two possible definitions of this notion. We also study the most important types of torsion theories, these being weight structures and t-structures (along with admissible triangulated subcategories). One of the main goals of the paper is to demonstrate that several important definitions and properties of weight structures and t-structures naturally extend to general torsion theories (so, we define smashing and cosmashing torsion theories); this observation leads to a certain optimization of proofs. Similarly we generalize the notion of adjacent and orthogonal weight and t-structures. Moreover, we relate adjacent torsion theories to the Brown-Comenetz duality and Serre functions; we hope to apply these results to the study of t-structures in compactly generated categories and in certain derived categories of coherent sheaves. Furthermore, we prove some completely new statements on the behaviour of torsion theories un