Thallium-Activated Sodium Iodide (NaI(Tl)) scintillators have been widely used for radiation detection since the middle of the XX th century, being applied, for instance, in nuclear medicine, environmental monitoring, nuclear physics, aerial survey, well logging, homeland security, etc. Among other remarkable features, NaI(Tl) offers a very high intrinsic scintillation light yield and ease of growing large-size crystals. On the other hand, the hygroscopic character of the material complicates the manipulation and requires a tight housing of the detector system preventing humidity from reaching the crystal. Energy ranges from a few keV to several MeV are accessible with state-of-the-art technology, using Photomultiplier Tubes (PMTs) for the light readout. These detectors have been successfully applied since the nineties of the XX th century in the direct search for dark matter in the form of hypothetical WIMPs (Weakly Interacting Massive Particles) pervading the galactic halo. WIMPs could explain the galactic dynamics and the 26% of the Universe matter-energy content required to explain the cosmic microwave background radiation anisotropies, and many other cosmological observations, within the standard ΛCDM cosmological model. DAMA/LIBRA experiment, taking data at Gran Sasso National Laboratory (LNGS), in Italy, and using NaI(Tl) detectors, has observed for more than twenty years an annual modulation in the detection rate. This modulation shares all the features expected for the galactic dark matter signal. However, no other experiment has observed any hint supporting the interpretation in terms of dark matter particle interactions of the DAMA/LIBRA result, which seems very difficult to reconcile with the plethora of negative results from different experiments (using different targets and techniques). Only very recently, three-sigma sensitivity to DAMA/LIBRA result is at hand using the same target material, NaI(Tl). This allows us to cancel all the signal dependencies on the particle dark matter model and the dark halo model, enabling a model independent evaluation of that result. In this article, experimental efforts using NaI(Tl) detectors aiming at testing the DAMA/LIBRA signal will be briefly revised, as well as the results released, data-taking status, possible systematics affecting this testing, and sensitivity prospects for the near future. Finally, some R&D efforts toward the development of new experimental approaches using either NaI(Tl) or undoped NaI crystals will be revised in the context of the solving of the DAMA/LIBRA puzzle on annual modulation but also moving forward to other possible applications.