Physicists seem to get huge computational value by introducing Grassman variables and Grassman integration into differential geometric calculations.

See: http://en.wikipedia.org/wiki/Grassmann_number

Can someone here motivate these techniques mathematically, and include the simplest pure-math example where their use and value can be illustrated. I have thought they were invented for computing volumes of constrained moduli spaces, although physicists did not originally describe them this way. Do any mathematicians use them rigorously to actually solve problems?