Semi-orthogonal decompositions (hereditary torsion pair, or Bousfield localizations) of the derived categories of abelian categories are used in algebraic geometry to study Fourier-Mukai transforms on derived categories of coherent sheaves, and in homotopy theory to get t-structures of triangulated categories. However, a fundamental question still remains: How to get such decompositions. In this talk, we shall provide conditions to construct such decompositions at the level of abelian categories. This applies to localizing subcategories, homological ring epimorphisms, commutative noetherian rings and non-singular rings.