This text presents the mathematical concepts of Grassmann variables and the method of
supersymmetry to a broad audience of physicists interested in applying these tools to
disordered and critical systems as well as related topics in statistical physics. Based on
many courses and seminars held by the author one of the pioneers in this field the reader is
given a systematic and tutorial introduction to the subject matter. The algebra and analysis of
Grassmann variables is presented in part I. The mathematics of these variables is applied to a
random matrix model path integrals for fermions dimer models and the Ising model in two
dimensions. Supermathematics - the use of commuting and anticommuting variables on an equal
footing - is the subject of part II. The properties of supervectors and supermatrices which
contain both commuting and Grassmann components are treated in great detail including the
derivation of integral theorems. In part III supersymmetric physical models are considered.
While supersymmetry was first introduced in elementary particle physics as exact symmetry
between bosons and fermions the formal introduction of anticommuting spacetime components can
be extended to problems of statistical physics and since it connects states with equal
energies has also found its way into quantum mechanics. Several models are considered in the
applications after which the representation of the random matrix model by the nonlinear
sigma-model the determination of the density of states and the level correlation are derived.
Eventually the mobility edge behavior is discussed and a short account of the ten symmetry
classes of disorder two-dimensional disordered models and superbosonization is given.
