Transparent conducting oxides (TCOs) and transparent oxide semiconductors (TOSs) have become necessary materials for a variety of applications in the information and energy technologies, ranging from transparent electrodes to active electronics components. Perovskite barium stannate (BaSnO3), a new TCO or TOS system, is a potential platform for realizing optoelectronic devices and observing novel electronic quantum states due to its high electron mobility, excellent thermal stability, high transparency, structural versatility, and flexible doping controllability. This article reviews recent progress in the doped BaSnO3 system, discussing the wide range of physical properties, electron-scattering mechanism, and demonstration of key semiconducting devices such as pn diodes and field-effect transistors. Moreover, we discuss the pathways to achieving two-dimensional electron gases at the interface between BaSnO3 and other perovskite oxides and describe remaining challenges for observing novel quantum phenomena at the heterointerface.