Applications
In quantum field theory, Grassmann numbers are the "classical analogues" of anticommuting operators. They are used to define the path integrals of fermionic fields. To this end it is necessary to define integrals over Grassmann variables, known as Berezin integrals.
Grassmann numbers are also important for the definition of supermanifolds (or superspace) where they serve as "anticommuting coordinates".
Read more about this topic: Grassmann Number