Experimentally it is shown that without any oxygen manipulation for TiO2, a strong room temperature ferromagnetism could be expected only in ultra-thin films, with the ideal thickness below 100 nm. Both bulks and nano-powders of TiO2 are diamagnetic, indicating that the surface and its nano-sublayers play very important roles in tailoring the magnetic properties in this type of compound. To shed a new light on the defect-related magnetism in the typical case of anatase TiO2 surfaces, we have performed a series of quantum-mechanical calculations for TiO2 slabs containing Ti or O vacancies in different distances from the (001) surface. The lowest formation energies were obtained for the Ti vacancies in the first sub-surface layer and the O vacancies within the surface. The computed magnetic states reflect complicated structural relaxations of atoms influenced by both the surface and vacant atomic positions. O atoms cannot contribute much to magnetic moment when Ti vacancies are isolated and far from the surface. Ti vacancies in TiO2 are only metastable. The formation energy of Ti interstitials is lower than for Ti vacancies since high-temperature annealing, especially with a lot of O2 available that would fill up O-related defects, and as a result, eliminate most of Ti vacancies. Lower temperatures, less O2, and shorter exposure times may enable not only partial elimination of Ti vacancies but also can facilitate their diffusion into different states of aggregations. In the ferromagnetic films (i.e. thin films below 100 nm), it looks like that the O atoms are located closer to the Ti vacancies.