The notion of an exact category has been introduced by Quillen to axiomatize the homological properties of extension-closed subcategories of abelian categories. It allows to define and study homological properties of an exact category, and to define its derived category. However, it turns out that the fundamental concept of length, as known for modules, is less suitable to be studied in the context of an exact category. We aim in this talk to present some recent developments showing for which kind of exact categories an analogue of the Jordan-Hölder property holds, and what one can expect from the notion of length in general. We also present results on the lattice structure of the set of all exact structures that can be attached to a fixed additive category.
Some of the presented results are joint work with Rose-Line Baillargeon, Mikhail Gorsky, Souheila Hassoun and Aran Tattar.