We review some of the techniques used to study the dynamics of disordered
systems subject to both quenched and fast (thermal) noise. Starting from the
Martin–Siggia–Rose/Janssen–De Dominicis–Peliti path integral formalism for
a single variable stochastic dynamics, we provide a pedagogical survey of the
perturbative, i.e. diagrammatic, approach to dynamics and how this formalism
can be used for studying soft spin models. We review the supersymmetric
formulation of the Langevin dynamics of these models and discuss the physical
implications of the supersymmetry. We also describe the key steps
involved in studying the disorder-averaged dynamics. Finally, we discuss the
path integral approach for the case of hard Ising spins and review some recent
developments in the dynamics of such kinetic Ising models.