In this chapter we will start with the definition of the so-called N-extended super- symmetric quantum mechanics. This approach, actually for N = 2, has first been formulated by Nicolai [Nic76] in his search for supersymmetry in non-relativistic quantimi systems related to models of statistical physics. Witten’s [Wit8l, Wit82a] approach has been for a general N ≥ 1 but, for N ≥ 2, has the same features as Nicolai’s formulation. Here we will present the definition due to Witten in order to include the N = 1 case, which has some interesting applications in the theory of a three-dimensional electron gas in a magnetic field. However, only for N ≥ 2 we will find the typical properties of supersymmetric quantum mechanics. In particular, N ≥2 allows the introduction of the so-called Witten parity and thus implies a natural grading of the Hilbert space into two subspaces.