Ghosts are a new class of approximate solutions to nonlinear boundary-value problems (BVPs). They are bounded away from true solutions, but solve an initial-value problem (IVP) for the differential equations exactly, and approximate boundary and internal matching conditions with exponentially decaying error as a control parameter approaches infinity. Ghosts are characteristic of singularly perturbed differential equations in which solutions are localized and the IVP depends sensitively on initial data. They correspond to shifted, localized solutions and they can be observed in (imperfect) physical experiments and be approximated by convergent discretization schemes. The latter property suggests that the concept of the 'numerically irrelevant' solution, introduced for discretized second-order BVPs, should be re-examined.