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 P21/c, with unit-cell parameters at 195 K of a = 10.26619 (3), b = 7.01113 (2), c = 10.06897 (3) Å, β = 101.5030 (2)° and V = 710.183 (3) Å3 (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 α phase is monoclinic (space group P21/c), with unit-cell parameters at 175 K of a = 6.30304 (4), b = 9.05700 (5), c = 11.22013 (7) Å, β = 105.9691 (4)° and V = 615.802 (4) Å3 (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, α-DMSO·2H2O undergoes a solid-state transformation to a mixture of DMSO·3H2O + anhydrous DMSO, and there is then a stable eutectic between these two phases at ∼203 K. The β-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 α-phase; above ∼180 K, β-DMSO·2H2O converts irreversibly to α-DMSO·2H2O. At 175 K, the lattice parameters of β-DMSO·2H2O are a = 6.17448 (10), b = 11.61635 (16), c = 8.66530 (12) Å, β = 101.663 (1)° and V = 608.684 (10) Å3 (Z = 4), hence this polymorph is just 1.16% denser than the α-phase under identical conditions. Like the other two hydrates, the space group appears likely, on the basis of systematic absences, to be P21/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).