We extend the model of ferromagnetic superexchange in dilute magnetic semiconductors to the ferromagnetically ordered highly insulating compounds (dilute magnetic dielectrics). The intrinsic ferromagnetism without free carriers is observed in oxygen-deficient films of anatase TiO 2 doped with transition metal impurities in cation sublattice. We suppose that ferromagnetic order arises due to superexchange between complexes [oxygen vacancies + magnetic impurities], which are stabilized by charge transfer from vacancies to impurities. The Hund rule controls the superexchange via empty vacancy related levels so that it becomes possible only for the parallel orientation of impurity magnetic moments. The percolation threshold for magnetic ordering is determined by the radius of vacancy levels, but the exchange mechanism does not require free carriers. The crucial role of the non-stoichiometry in formation of the ferromagnetism makes the Curie temperatures extremely sensitive to the methods of sample preparation. * Email:fleurov@post.tau.ac.il. Among the most salient features of magnetism in dilute ferromagnetic oxides one should mention an extreme sensitivity of the magnetic order to the growth and annealing conditions. 5, 8 We believe that this is an integral feature of magnetism in these materials, and the ferromagnetic ordering with high T C is mediated by intrinsic or extrinsic defects, which form complexes with magnetic dopants. Such point of view is supported by recent experimental studies of Co-doped 9 and Cr-doped 10 TiO 2 . It was noticed, in particular, that in the most perfect (Ti, Cr)O 2 crystals the hysteresis loop at a given temperature is essentially less distinct than in 'bad quality' samples. 10 Besides, the enormous scatter of effective magnetic moment per cation is observed in all ferromagnetic oxides, and the concentration of magnetic dopants lies far below the percolation threshold x c associated with the nearest-neighbor cation coupling. 8 Basing on these facts, one concludes that non-magnetic defects are also involved in formation of the localized magnetic moments. They influence their magnitude and localization, and this influence is sensitive to the growth and annealing regimes.Recent theories of magnetism in n-type dilute magnetic oxides appeal to magnetic polarons as mediators of indirect exchange between magnetic dopants. 8,11 In this case the extrinsic defects are donor impurities, which donate free electrons occupying the bottom of conduction band. These electrons form shallow spin-polarized polaronic states due to strong exchange with the localized magnetic moments of transition metal dopants. A large radius of these states makes the polaronic percolation threshold essentially lower than x c .One should note, however, that the carrier concentration in these compounds is vanishingly small, and that is why they have been proposed to be called dilute magnetic dielectrics (DMD). 9 It means that we need now to find an exchange mechanism, which works in the case of really insulating ox...