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Rhenium disulfide (ReS2), known in nature as the mineral rheniite, is a very interesting compound owing to its remarkable fundamental properties and great potential to develop novel device applications. Here we perform density functional theory (DFT) calculations to investigate the structural properties and compression behavior of this compound and also of the (Re,Mo)S-2 solid solution as a function of Re/Mo content. Our theoretical analysis is complemented with high-pressure X-ray diffraction (XRD) measurements, which have allowed us to reevaluate the phase transition pressure and equation of state of 1T-ReS2. We have observed the 1T-to-1T' phase transition at pressures as low as similar to 2 GPa, and we have obtained an experimental bulk modulus, B-0, equal to 46(2) GPa. This value is in good agreement with PBE+D3 calculations, thus confirming the ability of this functional to model the compression behavior of layered transition metal dichalcogenides, provided that van der Waals corrections are taken into account. Our experimental data and analysis confirm the important role played by van der Waals effects in the high-pressure properties of 1T-ReS2., J.I. and C.P. acknowledge the Spanish Ministry of Science, Innovation and Universities via Grant FIS2017-2017-83295-P. T.W. acknowledges financial support from Polish Ministry of Science and Higher Education via Diamond Grant no. D\2015 002645.