  We introduce the notion of composition series and the length of triangulated
categories, and we study compositions series of derived categories of certain
projective varieties and finite dimensional algebras. For example, we compute
the length of the derived category ${\rm D}^{\rm b}(C)$ of a smooth projective
curve $C$ and classify all finite length thick subcategories of ${\rm D}^{\rm
b}(C)$. Furthermore, we prove that the derived categories of certain smooth
projective varieties have composition series of different lengths. We also
discuss the length of (1) derived categories of finite dimensional
representations of Dynkin and extended Dynkin quivers, (2) derived categories
of some singular varieties and (3) Krah's phantom subcategories.
