The adsorption and reactivity of methanol on TiO2 anatase(101) surfaces are studied through first-principles total energy calculations and Car-Parrinello molecular dynamics simulations. The effects of different methanol coverages up to one monolayer and different surface oxidation states are investigated. The most stable monolayer state involves two different molecular adsorption states, with specific vibrational and electronic properties. Molecular adsorption is always favored on the stoichiometric surface, even though dissociation becomes increasingly more favorable with increasing coverage. Oxygen vacancies shift the energy balance in favor of the dissociated state, whereas dissociative and molecular adsorption appear competitive on the hydroxylated surface. On the latter, a specific molecular adsorption mode, not available on the clean surface, turns out to be stable. The vibrational and electronic properties of adsorbed methanol and methoxy are determined and compared to available experimental data.