The neutron Time-Of-Flight (n_TOF) facility at the European Laboratory for Particle Physics (CERN) is a pulsed white-spectrum neutron spallation source producing neutrons for two experimental areas: EAR1, located 185 m downstream of the spallation target, and EAR2, located 20 m above the target. The facility is based on a lead target impacted by a high-intensity 20 GeV/c proton beam. It is designed to study neutron-nucleus interactions for neutron kinetic energies from a few meV to several GeV, with applications in nuclear astrophysics, nuclear technology, and medical research. The facility is undergoing a major upgrade in 2019-2020, which will include the installation of the new third-generation target. The second-generation target consists in a water-cooled lead cylinder, while the new target will be cooled by nitrogen to avoid erosion-corrosion phenomena and contamination of the cooling water with radioactive lead spallation products. The new design will be optimized also for the vertical flight path. The operation of the new spallation target will start in 2021. This paper presents an overview on the evolution of the design and on the related R&D activities (including beam irradiation tests) carried out to ensure the best performance for both experimental areas and avoid the contamination issues of the previous targets.