GaN-based high electron mobility transistors (HEMTs) are promising for high speed and high power electronics applications. AlN and In0.17Al0.83N enable maximum possible vertical scaling of barriers in Ga-polar GaN-based HEMTs for high speed performance. High quality epitaxial growth of these barriers is a pre-requisite for superior device performance. In this work, plasma-assisted molecular beam epitaxy (MBE) was used to study the epitaxial growth of AlN/GaN and InAlN/GaN heterostructures. For a single AlN/GaN heterojunction (with sheet electron density of ~ 1.73x1013 cm-2) we measured Hall-effect mobility of ~ 1834/11460 cm2V-1s-1 (at room temperature/10 K). This is among the highest values reported in literature for single AlN/GaN heterojunctions and indicates the superior quality of these MBE grown films. For double AlN/GaN heterojunctions, we measured (under single channel approximation) very high mobility ~ 1890/14430 cm2V-1s-1 (at room temperature/10 K) and sheet electron density of ~ 1.87x1013 cm-2 at room temperature. This is the highest mobility reported to date for multiple AlN/GaN heterojunctions. Employing these films we demonstrated double-channel metal-oxide-semiconductor HEMTs (DC-MOSHEMT). This is the first report of AlN/GaN heterojunction based DC-MOSHEMT in literature to the best of our knowledge. By implementing top barrier recess and regrown contacts in future, these double barrier heterostructures may be employed to reduce DC - RF dispersion in III-nitride HEMTs.  To explore another novel approach of solving this problem, we investigated low temperature (LT-) MBE growth of AlN for in-situ surface passivation of HEMT structures. We found a window below 250 &#176;C to grow partially amorphous LT-AlN. For a LT-AlN capped AlN/GaN HEMT, we report a low DC-RF dispersion with gate lag and drain lag below 2%. Additionally, we investigated crystalline AlN growth at a low temperature of ~ 480 &#176;C. This was motivated by novel MBE grown AlN barrier HEMT structures with high mobility InGaN channels for Terahertz applications.  To fully exploit the potential of MBE grown high quality HEMT structures, low resistance (< 0.1 &#937;.mm) regrown source/drain ohmic contacts are essential. From a systematic study using MBE grown bulk n+ GaN films, effect of Si doping concentration on contact resistance was analyzed.