  We define the Segre numbers of an ideal as a generalization of the
multiplicity of an ideal of finite colength. We prove generalizations of
various theorems involving the multiplicity of an ideal such as a principle of
specialization of integral dependence, the Rees-B\"oger theorem, and the
formula for the multiplicity of the product of two ideals. These results are
applied to the study of various equisingularity conditions, such as Verdier's
condition W, and conditions $A_f$ and $W_f$.
