Summary: We give a survey of works (mostly for the last ten years) devoted to statements and solutions of parabolic problems modelling physical processes in solids having a discontinuity on the boundary at the initial instant of time. For one-dimensional processes, the notion of generalized vortex diffusion is introduced, which is characterized by rather general kinematics of the process, physical nonlinearity of the medium, and type of boundary condition at the point of discontinuity. We classify the cases where there exists self-similarity. A detailed analysis is given for non-Newtonian power-law fluid, for a medium, similar in properties to a rigidly-ideally-plastic solid, and also for the viscoplastic Shvedov-Bingham solid. For the latter case, we give a survey of numerically-analytic investigation methods of the Stefan problem.