Ion traps offer a way to determine nuclear binding energies through atomic mass measurements with a high accuracy and they are routinely used to provide isotopically or even isomerically pure beams of short-living ions for post-trap decay spectroscopy experiments. In this review, different ion-trapping techniques and progresses in recent nuclear physics experiments employing low-energy ion traps are discussed. The main focus in this review is on the benefit of recent high accuracy mass measurements to solve some key problems in physics related to nuclear structure, nuclear astrophysics as well as neutrinos. Also, several cases of decay spectroscopy experiments utilizing trap-purified ion samples are summarized.