The crystal Cs2LiYCl6:Ce (CLYC) is a very interesting scintillator material because of its good energy resolution and its capability to identify γ-rays and fast/thermal neutrons. The crystal Cs2LiYCl6:Ce contains 6Li and 35Cl isotopes, therefore, it is possible to detect thermal neutrons through the reaction 6Li(n, α)t while 35Cl ions allow to measure fast neutrons through the reactions 35Cl(n, p)35S and 35Cl(n, α)32P. In this work two CLYC 1″1″ crystals were used: the first crystal, enriched with 6Li at 95% (CLYC-6) is ideal for thermal neutron measurements while the second one, enriched with 7Li at >99% (CLYC-7) is suitable for fast neutron measurements. The response of CLYC scintillators was measured with different PMT models: timing or spectroscopic, with borosilicate glass or quartz window. The energy resolution, the neutron-γ discrimination and the internal activity are discussed. The capability of CLYC scintillators to discriminate γ rays from neutrons was tested with both thermal and fast neutrons. The thermal neutrons were measured with both detectors, using an AmBe source. The measurements of fast neutrons were performed at the Frascati Neutron Generator facility (Italy) where a deuterium beam was accelerated on a deuterium or on a tritium target, providing neutrons of 2.5 MeV or 14.1 MeV, respectively. The different sensitivity to thermal and fast neutrons of a CLYC-6 and of a CLYC-7 was additionally studied.