Over the past six years, superconductivity at high temperatures has been reported in a variety of hydrogen-rich compounds under high pressure. That high-temperature superconductivity should exist in these materials is expected according to the conventional theory of superconductivity, as shown by detailed calculations. However here we argue that experimental observations rule out conventional superconductivity in these materials. Our results indicate that either these materials are unconventional superconductors of a novel kind, which we term `nonstandard superconductors', or alternatively that they are not superconductors. If the former, we point out that the critical current in these materials should be five orders of magnitude larger than in standard superconductors, potentially opening up the way to important technological applications. If the latter, which we believe is more likely, we suggest that the signals interpreted as superconductivity are either experimental artifacts or they signal other interesting physics but not superconductivity.