I. A brief history of homological algebra. Homological algebra is the study of the methods of algebraic topology that have invaded extensively the domain of pure algebra during the last decades. The history of HA originated in the 19th century, through an initial aim at understanding an object, the so-called 'homology numbers', as appeared in the work of Riemann (1857) and Betti (1871), and a rigorous development by Poincaré (1895). Later in 1956, Cartan and Eilenberg wrote a book on 'Homological Algebra', which was a remarkable revolution in mathematics. As such, it led to a new beginning in the realm of the subject. One could have a bird's-eye view of the revolutionized theories on three periods: (1) HA associated with regular local rings influenced by Cartan and Eilenberg in the early 1960s, (2) HA involving Abelian categories and Sheaf cohomology inspired by Grothendieck and Serre continued through the 1980s, and (3) HA involving derived categories and triangulated categories, which is still ongoing to date. 

II. The purpose of the training school. The main aim of this school is to present the foundation of the three-period development, plus some discussion on its further studies.

III. A complementary flavor. We include talk shows in our training program, which takes place at the end of the program. Please see it in detail in the 'Speakers' section.