In this dissertation, we present band structure studies of three types of materials, which are wurtzite GaAs, atomically thin MoS2, and highly correlated rare-earth nitrides, by using the quasiparticle self-consistent GW (QSGW) method. First, we report results for wurtzite GaAs, which is found to be stable and coexists with the zinc-blende crystal structure (more stable in bulk) in nanowires. We provide detailed band structure parameters, such as effective mass parameters, band gap, crystal field splitting, spin-orbit splitting, etc., of wurtzite GaAs. This information on the bulk band structure is needed for the study of the nanowire specific electronic states, which can be obtained within the envelope function or effective mass type theories. The strain effects on the band structure parameters are also studied, because a nanowire could be under strain due to surface tension and tension caused by the matching of the lattice constant between wurtzite and zinc-blende sections in the wire. The band structure parameters of more well known zinc-blende GaAs was also calculated in order to test the validity of our QSGW calculations. We also present the band structure of 4H GaAs as a guideline of how the band parameters might change when there is mixing between hexagonal structure (wurtzite) and cubic structure (zinc blende) in the same nanowire. Second, we present the results of bulk and atomically thin MoS2. Our QSGW results confirm the transition of the band gap nature from indirect gap to direct gap when the form of MoS2 changes from bulk to monolayer. However, the QSGW significantly overestimates the direct gaps at the K point of monolayer and bilayer MoS2 due to the very strong excitonic effect in this two dimensional material, which is not taken into account in the QSGW method. Therefore, we also estimated the exitonic effect by using the Mott-Wannier effective mass theory, and obtained a large ground state exciton binding energy for both monolayer and bilayer. Our final results of the transitions at the K point are in the very good agreement with the photoluminescence measurements. In addition, we also study the strain effect on the spin-orbit induced band structure splittings in the monolayer MoS2 and graphene, which provide an important piece of information for the study of spin scattering in these materials. Finally, the band structure studies of rare-earth nitrides, such as GdN, DyN, SmN, HoN, and YbN are presented. This group of materials is challenging in terms of the band structure calculations because of the highly correlated nature of open f shells. We found from our QSGW results that band splittings near the valence band maximum of these compounds are quite unique because they strongly depend on how they interact with the occupied 4f levels. We also show how good QSGW predicts the levels of the 4f states and the magnetic moments when compared to the experimental data.