Characteristic classes of hypersurfaces and characteristic cycles

We give a new formula for the Chern-Schwartz-MacPherson class of a hypersurface with arbitrary singularities, generalizing the main result of [P-P), which was a formula for the Euler characteristic. Two different approaches are presented. The first is based on the theory of characteristic cycles of a D-module (or a holonomic system) and the work of Sabbah [S], Briancn-Maisonobe-Merle [B-M-M], and Le-Mebkhout [L-M]. In particular, this approach leads to a simple proof of a formula of Aluffi [A] for the above mentioned class. The second approach uses Verdier's [V] specialization property of the Chern-Schwartz-MacPherson classes. Some related new formulas for complexes of nearby cycles and vanishing cycles are also given.