The final analysis of atmospheric neutrino events collected with the MACRO detector is presented. Three different classes of events, generated by neutrinos in different energy ranges, are studied looking at rates, angular distributions and estimated energies. The results are consistent for all the subsamples and indicate a flux deficit that depends on energy and path-length of neutrinos. The no-oscillation hypothesis is excluded at ∼ 5σ, while the hypothesis of vμ → vτ oscillation gives a satisfactory description of all data. The parameters with highest probability in a two flavor scenario are sin2 2θm = 1 and Δm2 = 0.0023 eV2. This result is independent of the absolute normalization of the atmospheric neutrino fluxes. The data can also be used to put experimental constrain on this normalization.