  In this paper we give an introduction to our recent work on characteristic
classes of complex hypersurfaces based on some talks given at conferences in
Strasbourg, Oberwolfach and Kagoshima. We explain the relation between nearby
cycles for constructible functions or sheaves as well as for (relative)
Grothendieck groups of algebraic varieties and mixed Hodge modules, and the
specialization of characteristic classes of singular spaces like the Chern-,
Todd-, Hirzebruch- and motivic Chern-classes. As an application we get a
description of the differences between the corresponding virtual and functorial
characteristic classes of complex hypersurfaces in terms of vanishing cycles
related to the singularities of the hypersurface.
