We report the electrical and optical properties of undoped and Mg-doped InN grown by molecular beam epitaxy on sapphire (0 0 0 1) substrates. InN has a hexagonal structure and its c-axis is normal to the (0 0 0 1) sapphire surface. The Raman spectra show a strong E-2 (low) mode at 87 cm(-1). The band gap energies of the samples are much smaller than 1.9 eV. InN with the band gap energy less than 1.5 eV shows resistivity behavior anomaly below 2.5 K. The relation between the low temperature resistivity anomaly and the band gap energy of InN is discussed.