In this work, high quality InN films grown on GaN substrates using RF plasma assisted Molecular Beam Epitaxy (MBE) are reported and the structural, electrical and optical characterization are presented. Structural quality was characterized by high resolution X-ray diffraction and transmission electron microscopy to optimize the growth conditions. The transport properties of InN have been studied using Hall effect measurements. A close correlation is found between the structural quality and the electron transport properties of InN. Magnetic field dependent Hall effect measurements were performed and a quantitative mobility spectrum analysis (QMSA) was used to extract the mobility spectrum for electrons in InN. With photoluminescence and absorption spectroscopy measurements, the optical band gap of InN was found to be ~ 0.67 eV. To study the conduction band offset between InN and GaN, n-n isotype InN/GaN heterojunction diodes were grown and fabricated. The devices were characterized by I-V, C-V, and photocurrent spectroscopy measurements. The conduction band offset between InN and GaN was extracted to be ~ 1.67+/- 0.1 eV. An AlN/GaN HEMT using InN as the gate was fabricated. The study showed that the InN gate only has weak modulation in source/drain current due to an ultra thin AlN barrier layer. This research outlined a path for the future pursuit of utilizing InN for HEMT applications. To explore the application of InN in solar cells, a p-GaN/InN-GaN heterojunction diode was grown and fabricated. The device showed photocurrents in response to visible lights, and also exhibited a photovoltaic effect. This study indicates the feasibility of incorporating InN and high In containing InGaN in traditional nitrides, and outlines a clear pathway for further investigation in this area.