Protonated and deuterated samples of the hydrogensulfide of cesium were studied by high-resolution neutron powder diffraction, calorimetry and proton NMR investigations in a wide temperature range. Primarily due to reorientational disorder of the anions, three modifications of the title compounds are known: an ordered low-temperature modification-LTM (tetragonal, I4/m, Z=8), a dynamically disordered middle-temperature modification-MTM (tetragonal, P4/mbm, Z=2), and a high-temperature modification-HTM (cubic, Pm (3) over barm, Z=1). The LTMreversible arrowMTM phase transition is continuous. Its order parameter, related to an order/disorder and to a displacive part of the phase transition, coupled bilinearly, follows a critical law. The critical temperature T-C=123.2+/-0.5 K determined by neutron diffraction of CsDS is in good agreement with T-C=121+/-2 K obtained by calorimetric investigations. For the protonated title compound a shift to T-C=129+/-2 K was observed by calorimetric measurements. The entropy change of this transition is (0.24+/-0.04) R and (0.27+/-0.04) R for CsHS and CsDS, respectively. The MTMreversible arrowHTM phase transition is clearly of first order. The transition temperatures of CsHS and CsDS are T=207.9+/-0.3 K and T=213.6+/-0.3 K with entropy changes of (0.86+/-0.01) R and (0.81+/-0.01) R, respectively. Second moments (M-2) of the proton NMR absorption signal of MTM and HTM are in reasonable agreement with M-2 calculated for the known crystal structures. A minimum in spin-lattice relaxation times (T-1) in the MTM could not be assigned by dipolar coupling to a two-site 180degrees reorientation of the anions, a model of motion presumed by the knowledge of the crystal structure. The activation enthalpies determined by fits of T-1 presuming a thermal activated process are in the order of molecular reorientations (E-a=13.5+/-0.5 kJ mol(-1) for the MTM and E-a=9.3+/-0.3 kJ mol(-1) for the HTM). In the HTM at T>330 K translational motion of the cations determines T-1 (E-a=13.8+/-0.4 kJ mol(-1)).