The epsilon multiplicity of a module is a natural generalization of the Hilbert-Samuel multiplicity of an ideal. It can be expressed as the volume of a suitable Cartier divisor associated with the module. In this talk, I will present a result that determines the change of the epsilon multiplicity of a module across flat families. The change is the multiplicity of the polar curve associated with the module. This result generalizes previous work of Gaffney, Kleiman, Teissier and Hironaka. I will discuss various applications of this relation, among which to the Whitney-Thom equisingularity theory and the topology of isolated singularities.