The x-ray structure factor of molten TiO2 has been measured, enabled by the use of aerodynamic levitation and laser beam heating, to a temperature of T=2250(30)K. The Ti-O coordination number in the melt is close to nTiO=5.0(2), with modal Ti-O bond length rTiO=1.881(5)Å, both values being significantly smaller than for the high temperature stable rutile crystal structure (nTiO=6.0,rTiO=1.959Å). The structural differences between melt and crystal are qualitatively similar to those for alumina, which is rationalized in terms of the similar field strengths of Ti4+ and Al3+. The diffraction data are used to generate physically and chemically reasonable structural models, which are then compared to the predictions based on various classical molecular dynamics (MD) potentials. Interatomic potentials, suitable for modeling molten TiO2, are introduced, given the inability of existing MD models to reproduce the diffraction data. These potentials have the additional advantage of being able to predict the density and thermal expansion of the melt, as well as solid amorphous TiO2, in agreement with published results. This is of critical importance given the strong correlation between density and structural parameters such as nTiO. The large thermal expansion of the melt is associated with weakly temperature dependent structural changes, whereby simulations show that nTiO=5.85(2)-[3.0(1)×10-4]T(K ,2.75Åcutoff). The TiO2 liquid is structurally analogous to the geophysically relevant high pressure liquid silica system at around 27 GPa. We argue that the predominance of fivefold polyhedra in the melt implies the existence of as-yet-undiscovered TiO2 polymorphs, based on lower-than-octahedral coordination numbers, which are likely to be metastable under ambient conditions. Given the industrial importance of titanium oxides, experimental and computational searches for such polymorphs are well warranted.