The transport properties of high mobility AlGaN/AlN/GaN and high sheet electron density AlInN/AlN/GaN two-dimensional electron gas (2DEG) heterostructures were studied. The samples were grown by metal-organic chemical vapor deposition on c-plane sapphire substrates. The room temperature electron mobility was measured as 1700 cm(2)/V s along with 8.44 X 10(12) cm(-2) electron density, which resulted in a two-dimensional sheet resistance of 435 Omega/square for the Al(0.2)Ga(0.8)N/AlN/GaN heterostructure. The sample designed with an Al(0.88)In(0.12)N barrier exhibited very high sheet electron density of 4.23 X 10(13) cm(-2) with a corresponding electron mobility of 812 cm(2)/V s at room temperature. A record two-dimensional sheet resistance of 182 Omega/square was obtained in the respective sample. In order to understand the observed transport properties, various scattering mechanisms such as acoustic and optical phonons, interface roughness, and alloy disordering were included in the theoretical model that was applied to the temperature dependent mobility data. It was found that the interface roughness scattering in turn reduces the room temperature mobility of the Al(0.88)In(0.12)N/AlN/GaN heterostructure. The observed high 2DEG density was attributed to the larger polarization fields that exist in the sample with an Al0.88In0.12N barrier layer. From these analyses, it can be argued that the AlInN/AlN/GaN high electron mobility transistors (HEMTs), after further optimization of the growth and design parameters, could show better transistor performance compared to AlGaN/AlN/GaN based HEMTs. c 2009 American Institute of Physics.