  We give a new formula for the Chern-Schwartz-MacPherson class of a
hypersurface in a nonsigular compact complex analytic variety. In particular
this formula generalizes our previous result on the Euler characteristic of
such a hypersurface. Two different approaches are presented. The first is based
on the theory of characteristic cycle and the works of Sabbah,
Briancon-Maisonobe-Merle, and Le-Mebkhout. In particular, this approach leads
to a simple proof of a formula of Aluffi for the above mentioned class. The
second approach uses Verdier's specialization property of the
Chern-Schwartz-MacPherson classes. Some related new formulas are also given.
