Gravitational waves induced by primordial density perturbations provide a powerful probe of the Universe’s thermal history, which may include an early matter-dominated (eMD) era predicted by well-motivated particle-physics models. The induced GWs can be significantly enhanced when the Universe undergoes a sudden transition from an eMD era to an era with pressure, such as a radiation or kination era. This enhancement arises from the growth of density perturbations during the eMD era and their rapid oscillations during the era with pressure. This phenomenon is called the poltergeist mechanism. In this review, we explain the essence of the poltergeist mechanism and explore concrete scenarios in which such an enhancement can occur.