Exceptional collections in derived categories of coherent sheaves
have a long history going back to the pioneering work of A. Beilinson.
After recalling the general setup, we will concentrate on some recent
developments inspired by the homological mirror symmetry.
Namely, we will define residual categories of Lefschetz exceptional
collections and discuss a conjectural relation between the structure
of quantum cohomology and residual categories. We will illustrate this
relationship in the case of some isotropic Grassmannians. Based on joint
works with Alexander Kuznetsov.