The Damage Spreading (DS) method allows the investigation of the effect caused by tiny perturbations, in the initial conditions of physical systems, on their final stationary or equilibrium states. The damage (D(t)) is determined during the dynamic evolution of a physical system and measures the time dependence of the difference between a reference (unperturbed) configuration and an initially perturbed one. In this paper we first give a brief overview of Monte Carlo simulation results obtained by applying the DS method. Different model systems under study often exhibit a transition between a state where the damage becomes healed (the frozen phase) and a regime where the damage spreads arriving at a finite (stationary) value (the damaged phase), when a control parameter is finely tuned. These kinds of transitions are actually true irreversible phase transitions themselves, and the issue of their universality class is also discussed. Subsequently, the attention is focused on the propagation of damage in magnetic systems placed in confined geometries. The influence of interfaces between magnetic domains of different orientation on the spreading of the perturbation is also discussed, showing that the presence of interfaces enhances the propagation of the damage. Furthermore, the critical transition between propagation and nonpropagation of the damage is discussed. In all cases, the determined critical exponents suggest that the DS transition does not belong to the universality class of Directed Percolation, unlike many other systems exhibiting irreversible phase transitions. This result reflects the dramatic influence of interfaces on the propagation of perturbations in magnetic systems.