This talk is devoted to a conceptional introduction of the nonlinear stability theory of periodic wave trains. The main purpose is to explain that they are not only nonlinearly stable against co-periodic or localized perturbations, but also against combined and even fully nonlocalized perturbations. Such extensions are discussed for paradigm models of dissipative pattern-forming systems including the FitzHugh-Nagumo system, reaction-diffusion systems and the Lugiato-Lefever equation. We illustrate several ideas along simpler toy problems.