Group-III sesquioxides exhibit a pronounced polymorphism and, specifically In 2 O 3 and Ga 2 O 3 form wide-band-gap (WBG) and even ultra-wide-band-gap (UWBG) semiconductors (in the case of Ga 2 O 3 ) with promising properties for applications in optoelectronics and power electronics. Energy gaps ranging from 2.3 to 3.7 (In 2 O 3 ) or from 4.4 to 5.3 eV (Ga 2 O 3 ) with a strong absorption onset at around 3.6 eV (In 2 O 3 ) make these materials transparent in the ultraviolet (UV) or visible wavelengths region. Together with n-doping and high electrical conductivity, this unusual combination of high optical transparency and electrical conductivity places them in a very special class of materials of interest called transparent conducting oxides (TCOs). Five polymorphs of Ga 2 O 3 comprising rhombohedral corundum (α), monoclinic (β), defective spinel (γ), body-centered cubic bixbyite (δ), and orthorhombic or hexagonal (ε) phases are reported. Recently, a subgroup κ of the ε polymorph was discovered. This differentiation was accompanied by the reinterpretation of the originally identified orthorhombic structure to be a hexagonal structure, while the κ subgroup has orthorhombic symmetry. Similarly, pronounced polymorphism is also reported for indium oxide. It is possible to grow high-quality single-crystalline In 2 O 3 in the stable body-centered cubic (bcc) bixbyite phase, named bcc-In 2 O 3 or δ-In 2 O 3 . The existence of a high-pressure corundum-type metastable rhombohedral (rh) polymorph known as rh-In 2 O 3 or α-In 2 O 3 is also documented. Moreover, the orthorhombic phases ω1, ω2, and ω3-In 2 O 3 have also been discussed in literature. To explore and understand the potential of Ga 2 O 3 and In 2 O 3 compounds for possible applications, it is crucial to thoroughly examine their electronic structures and the resulting optical properties across the crystal structures of the polymorphs. Particular attention should be given to the tuning of the band gap, the corresponding dipole strength of the optical transitions, and the influence of excitonic effects. The relatively flat, oxygen-derived uppermost valence bands lead to direct and indirect gaps with similar values. Among these materials, β-Ga 2 O 3 has been extensively studied using various methods. Its optical gap values, ranging between 4.48-5.68 eV depend on the sample growth and the light polarization. Typical absorption edges have been found at 4.48, 4.57, or 4.70 eV for light polarized parallel to one of the crystallographic axes in transmittance and reflectance spectroscopy studies on bulk β-Ga 2 O 3 . Ellipsometry measurements typically show higher transition energies, such as 5.15, 5.37, and 5.68 eV 26 or 5.04, 5.40, and 5.64 eV, but they confirm the pronounced optical anisotropy. The fundamental gap not only depends on the optical method applied but also varies with the polymorph, although fewer experimental studies are available for the latter. For α-Ga 2 O 3 optical investigations suggest a band gap of approximately 5.3 eV. Ellipsometry studies reveal optical anisotropy with a low-energy M 1 van Hove singularity close to 5.46 eV. Slightly higher energies have been observed for epitaxial α-Ga 2 O 3 . The reported band gaps for ε(κ)-Ga 2 O 3 range from 4.4 -4.7 eV. Optical absorbance points toward a value of 4.51 eV. Using the same method, a fundamental gap of 5.0 eV is suggested for orthorhombic κ-Ga 2 O 3 . For In 2 O 3 , the experimental data for the fundamental gap vary widely, ranging from 2.72 -3.75 eV depending on the sample preparation, measurement method, and polymorph. The optical gaps derived for both bixbyite and rhombohedral In 2 O 3 tend to be around 3.0 eV. However, somewhat larger gaps have also been detected by spectroscopic ellipsometry. This trend is further confirmed by X-ray absorption and emission measurements. Theoretical investigations into the electronic band structure and fundamental gaps of gallium and indium oxides are challenging due to the necessity of incorporating many-body effects, such as QP (at the single-particle level) and excitonic (at the two-particle level) interactions, to accurately account for the excitation of electrons and holes as well as their interactions. Additionally, these calculations must often handle a large number of atoms-up to 40 or more-in the primitive unit cell. The many-body studies are additionally hampered due to the fact that the electronic screening of excited particles and their interaction is modified by the vibrating ionic lattice. Such rather complete calculations based on Hedin's GW approximation for the single-particle excitations and the solution of the Bethe-Salpeter equation (BSE) to account for excitonic effects in electron-hole pair excitations and, hence, optical gaps, are rare. Such studies are mainly restricted to Ga 2 O 3 and In 2 O 3 polymorphs with only 10 atoms in the primitive unit cell. GW calculations have been performed with different complexity and approximations for self-consistency and its starting point. The results vary between 4.62-5.30 eV for β-Ga 2 O 3 or 2.60-3.35 eV for cubic In 2 O 3 . To reduce computational effort and address polymorphs with more atoms in the unit cell, approximate methods have also been employed to account for QP excitation effects mediated by the exchange-correlation (XC) self-energy. Hybrid XC functionals are frequently used to overcome the underestimation of the fundamental gap observed in density functional theory (DFT), particularly in the electronic ground state when using local or semi-local XC functionals. A significant example is the HSE06 functional, a variant that incorporates screened Fock exchange according to Heyd, Scuseria, and Ernzerhof. However, the resulting gaps tend to be smaller than those derived from GW calculations. To account for excitonic effects from first principles, the BSE has also been solved for both Ga 2 O 3 and In 2 O 3 , particularly in polymorphs with a small number of atoms in the primitive unit cell. Theoretical results indicate that the filled Ga 3d /In 4d orbitals hybridize with the O 2s and O 2p orbitals. Additionally, there is a contribution from the Ga 3d /In 4d bands to the O 2p states in the valence band. DFT combined with pseudopotentials lacks the precision needed for accurately representing these localized states. The simplification introduced by pseudopotentials, treating such states as valence states, may not be sufficient for accurately describing them and capturing their effects. As a result, without QP effects, an incorrect description of the electronic structure and an inadequate representation of the effects of these states on the valence band maximum (VBM) occurs, thereby impacting the calculated energy band gap.