A superconductor is a material that can conduct electricity without resistance below a superconducting transition temperature, Tc. The highest Tc that has been achieved to date is in the copper oxide system1: 133 kelvin at ambient pressure2 and 164 kelvin at high pressures3. As the nature of superconductivity in these materials is still not fully understood (they are not conventional superconductors), the prospects for achieving still higher transition temperatures by this route are not clear. In contrast, the Bardeen–Cooper–Schrieffer theory of conventional superconductivity gives a guide for achieving high Tc with no theoretical upper bound—all that is needed is a favourable combination of high-frequency phonons, strong electron–phonon coupling, and a high density of states4. These conditions can in principle be fulfilled for metallic hydrogen and covalent compounds dominated by hydrogen5, 6, as hydrogen atoms provide the necessary high-frequency phonon modes as well as the strong electron–phonon coupling. Numerous calculations support this idea and have predicted transition temperatures in the range 50–235 kelvin for many hydrides7, but only a moderate Tc of 17 kelvin has been observed experimentally8. Here we investigate sulfur hydride9, where a Tc of 80 kelvin has been predicted10. We find that this system transforms to a metal at a pressure of approximately 90 gigapascals. On cooling, we see signatures of superconductivity: a sharp drop of the resistivity to zero and a decrease of the transition temperature with magnetic field, with magnetic susceptibility measurements confirming a Tc of 203 kelvin. Moreover, a pronounced isotope shift of Tc in sulfur deuteride is suggestive of an electron–phonon mechanism of superconductivity that is consistent with the Bardeen–Cooper–Schrieffer scenario. We argue that the phase responsible for high-Tc superconductivity in this system is likely to be H3S, formed from H2S by decomposition under pressure. These findings raise hope for the prospects for achieving room-temperature superconductivity in other hydrogen-based materials.