Muonium, and by analogy hydrogen, is shown to form a shallow-donor state in ${\text{In}}_{2}{\text{O}}_{3}$ and ${\text{SnO}}_{2}$. The paramagnetic charge state is stable below $$∼${}50\text{ }\text{K}$ in ${\text{In}}_{2}{\text{O}}_{3}$ and $$∼${}30\text{ }\text{K}$ in ${\text{SnO}}_{2}$ which, coupled with its extremely small effective hyperfine splitting in both cases, allows its identification as the shallow-donor state. This has important implications for the controversial issue of the origins of conductivity in transparent conducting oxides.