MACRO results on atmospheric neutrinos

We discuss the final results of the MACRO experiment on atmospheric neutrino oscillations. The data concern event topologies with average neutrino energies of ∼ 3 and ∼ 50 GeV. Multiple Coulomb Scattering of the high energy muons was used to estimate the neutrino energy event by event. The angular distributions, the L/Eν distribution, the particle ratios and the absolute fluxes all favor νμ → ντ oscillations with maximal mixing and ∆m ≃ 0.0023 eV. Emphasis is given to measured ratios which are not affected by Monte Carlo (MC) absolute normalization; a discussion is made on MC uncertainties. A preliminary search for possible Lorentz invariance violation contributions to atmospheric neutrino oscillations is presented and discussed.