Neutron powder diffraction data have been collected from a series of flash-frozen aqueous solutions of dimethyl sulfoxide (DMSO) with concentrations between 25 and 66.7 mol% DMSO. These reveal the existence of three stoichiometric hydrates, which crystallize on warming between 175 and 195 K. DMSO trihydrate crystallizes in the monoclinic space group P2(1)/c, with unit-cell parameters at 195 K of a = 10.26619 (3), b = 7.01113 (2), c = 10.06897 (3) angstrom, beta = 101.5030 (2)degrees and V = 710.183 (3) angstrom(3) (Z = 4). Two of the symmetry-inequivalent water molecules form a sheet of tiled four- and eight-sided rings; the DMSO molecules are sandwiched between these sheets and linked along the b axis by the third water molecule to generate water-DMSO-water tapes. Two different polymorphs of DMSO dihydrate have been identified. The alpha phase is monoclinic (space group P2(1)/c), with unit-cell parameters at 175 K of a = 6.30304 (4), b = 9.05700 (5), c = 11.22013 (7) angstrom, beta = 105.9691 (4)degrees and V = 615.802 (4) angstrom(3) (Z = 4). Its structure contains water-DMSO-water chains, but these are polymerized in such a manner as to form sheets of reniform eight-sided rings, with the methyl groups extending on either side of the sheet. On warming above 198 K, alpha-DMSO center dot 2H(2)O undergoes a solid-state transformation to a mixture of DMSO center dot 3H(2)O + anhydrous DMSO, and there is then a stable eutectic between these two phases at similar to 203 K. The beta-phase of DMSO dihydrate has been observed in a rapidly frozen eutectic melt and in very DMSO-rich mixtures. It is observed to be unstable with respect to the alpha-phase; above similar to 180 K, beta-DMSO center dot 2H(2)O converts irreversibly to alpha-DMSO center dot 2H(2)O. At 175 K, the lattice parameters of beta-DMSO center dot 2H(2)O are a = 6.17448 (10), b = 11.61635 (16), c = 8.66530 (12) angstrom, beta = 101.663 (1)degrees and V = 608.684 (10) angstrom(3) (Z = 4), hence this polymorph is just 1.16% denser than the alpha-phase under identical conditions. Like the other two hydrates, the space group appears likely, on the basis of systematic absences, to be P2(1)/c, but the structure has not yet been determined. Our results reconcile 60 years of contradictory interpretations of the phase relations in the binary DMSO-water system, particularly between mole fractions of 0.25-0.50, and confirm empirical and theoretical studies of the liquid structure around the eutectic composition (33.33 mol% DMSO).