This paper reviews progress in the heteroepitaxial growth of III-nitrides by plasma-assisted MBE. The role of nucleation layers in determining the polarity of these films is addressed. Doping of GaN with silicon leads to n-type films whose carrier concentration can be controlled between 10(16) and 10(20) cm(-3). P-type doping with Mg with free carrier concentration of up to 5 x 10(18) cm(-3) has been attained without requiring post-growth annealing. The epitaxial growth of AlGaN and InGaN alloys is addressed. AlGaN alloys were shown to undergo either one monolayer or seven and 12 monolayer superlattice ordering along the [0 0 0 1 6] direction, which is also the direction of growth. The InGaN alloys were found to undergo both ordering as well as phase separation. InGaN/GaN and GaN/AlGaN MQWs have been fabricated and found to show good radiative recombination efficiency. Carrier transport in lightly doped GaN films is anisotropic in the directions parallel and perpendicular to the substrate. In the parallel direction the mobility is reduced by dislocations while in the vertical direction the mobility attains values about 1000cm(2)/Vs, close to the theoretical prediction. AlN/GaN distributed Bragg reflectors (DBRs) with reflectivity 99% have been fabricated.