Over the past 20 years, III-nitrides (GaN, AlN, InN and their alloys) have proven to be an excellent material group for electronic devices, in particular, for high electron mobility transistors (HEMTs) operating at high frequency and high power. This is mainly thanks to the wide band gap of III-nitrides and the spontaneous polarization along the technologically relevant (0001) direction. Heteroepitaxy of III-nitride heterostructures enables the formation of two-dimensional electron gases (2DEGs) with high carrier densities (> 10^13 cm^-2) and high electron mobilities (2000 cm^2V^-1s^-1). The heterostructures usually consist of AlGaN, InAlN or AlN barriers grown epitaxially on GaN buffers. Devices based on these structures exhibit excellent performance and have reached technological maturity, enabling notably the commercialization of AlGaN/GaN HEMTs. Despite the success of III-nitride HEMTs thus far, the full potential of the material group for electronics has not yet been unleashed. The aim of this thesis is to investigate two novel heterostructure designs and to determine the physical mechanisms limiting their electronic properties. This was done by epitaxial growth followed by characterization of material and electronic properties. The first studied heterostructure was AlN/GaN/AlN where the GaN channel is fully strained to AlN. In order to achieve pseudomorphic growth of GaN on AlN, the onset of strain relaxation, i.e., the critical thickness, is of paramount importance. Furthermore, the impact of growth parameters such as temperature and initial dislocation density need to be considered. Over the past years several, research groups have reported on the electronic properties of ultra-thin GaN channels on AlN. Interestingly, all reported 2DEGs suffer from a low electron mobility with values usually below 600 cm^2V^-1s^-1. In this thesis, the strain relaxation and critical thickness of GaN on AlN were determined. Furthermore, the origin of low electron mobility for AlN/GaN/AlN heterostructures and in general thin-GaN channels grown on AlN were systematically studied. The second heterostructure that was investigated during this thesis was based on InGaN channels. In this case, the 2DEG is formed in an In-rich InGaN layer. The low electron effective mass of InN (0.05 m0) compared to GaN (0.2 m0) should theoretically give rise to a higher electron mobility. However, for In-rich channels alloy disorder scattering is particularity strong in III-nitrides. This has been shown both by optical measurements and theoretical calculations based on the localization landscape theory. Surprisingly, high electron mobilities have been reported in literature for high In content InGaN channels. In these cases, the electron mobility appears not to be limited by alloy disorder scattering. During this thesis, the electron mobility of InGaN channels was determined as a function of In content. Furthermore, the unintentional growth of GaN interlayers is proposed as the possible origin for the reported high electron mobilities in literature.