A host of spatially extended systems, both in physics and in other disciplines, are well described at a coarse-grained scale by a Langevin equation with multiplicative-noise. Such systems may exhibit nonequilibrium phase transitions, which can be classified into universality classes. Here we study in detail one such class that can be mapped into a Kardar-Parisi-Zhang (KPZ) interface equation with a positive (negative) non-linearity in the presence of a bounding lower (upper) wall. The wall limits the possible values taken by the height variable, introducing a lower (upper) cut-off, and induces a phase transition between a pinned (active) and a depinned (absorbing) phase. This transition is studied here using mean field and field theoretical arguments, as well as from a numerical point of view. Its main properties and critical features, as well as some challenging theoretical difficulties, are reported. The differences with other multiplicative noise and bounded-KPZ universality classes are stressed, and the effects caused by the introduction of "attractive" walls, relevant in some physical contexts, are also analyzed.