This paper describes studies on high-quality InN growth on sapphire by RF-MBE. Critical procedures to obtain high-quality InN films were investigated and (1) nitridation process of sapphire substrates prior to growth, (2) precise control of V/III ratio and (3) selection of optimum growth temperature were found to be essential. Detailed structural characterizations by XRD, TEM, Raman scattering and EXAFS indicate that InN films obtained in this study have ideal hexagonal wurtzite structure. FWHMs of ω-2Θ mode XRD and E2(high)-phonon-mode of Raman scattering are as small as 28.9 arcsec and 3.2 cm-1, respectively. True band gap energy of InN is also discussed based on optical characterization results obtained from well-characterized hexagonal InN grown in this study. PbS, instead of InGaAs, was used as a detector for PL study in order to solve the problem coming from the cut-off wavelength of InGaAs detector. Based on these systematic studies on structural and optical property characterizations using high-quality InN, true band-gap energy of InN is suggested to be less than 0.67 eV and approximately 0.65 eV at room temperature. Single-crystalline InN films are also successfully grown on Si substrates by a brief nitridation of the Si substrates. Significant improvement of InN crystal quality on Si substrates by the insertion of an AlN buffer layer is also demonstrated.