The emergence of ultra-wide-band-gap semiconductors and their deployment in high-power and high-frequency electronic devices has challenged the conventional distinction between semiconductors and insulators based solely on the magnitude of their band gap. It also raised fundamental questions about the extreme upper bound of band gaps that are compatible with semiconducting behavior. I will discuss the development and applications of computational discovery strategies based on first-principles electronic-structure and defect calculations to identify materials that combine ultra-wide band gaps with shallow dopants and mobile carriers. I will first discuss the discovery of semiconducting rutile GeO₂ and its alloys with SnO₂, which have since attracted experimental interest for applications in power electronics and UV transparent conductors. Despite their ultra-wide gaps (3.6–4.7 eV), our calculations predict that these materials exhibit shallow dopants, high mobilities, and high thermal conductivities. The n-type dopability has since been verified experimentally, and early transistor devices have been demonstrated. Surprisingly, both experiments and theory find that alloy disorder in Sn-rich (Ge,Sn)O₂ alloys has little impact on the electron mobility. We attribute this property to the insensitivity of the conduction-band edge with respect to alloy composition in the Sn-rich range, which suppresses spatial fluctuations of the electron energies.  We then apply this materials-discovery strategy to identify new semiconductors with band gaps exceeding that of AlN (6.2 eV) while retaining essential semiconducting properties such as shallow dopants and mobile charge carriers. We find that compounds composed of light elements in dense crystal structures such as tetrahedral BeO and rutile SiO₂ can sustain shallow dopants, weak polaron binding, and mobile carriers even for gaps as wide as 9.5 eV. Our findings demonstrate that semiconducting behavior persists even at extreme band gaps, far beyond conventional upper bounds traditionally associated with semiconductor materials.