The critical current density of a superconductor as a function of temperature and magnetic field is a key quantity for the design of superconductor applications. Some superconducting devices, e.g. fault current limiters, operate into the supercritical current range by design and benefit from the understanding of the superconducting transition in high current densities. Tremendous efforts are currently undertaken to enhance the critical current in technical materials by improving the pinning forces but are limited by the superconducting depairing current that usually cannot be measured directly. Only few experiments have explored the supercritical current region so far. We report on such measurements in extremely high current densities. In order to avoid thermal heating and eventually destruction of our samples that results inevitably from such high dissipation, the measurements are carried out using short-pulse currents with about 50 ns duration. In addition, we use ultrathin films of YBCO, patterned to microbridges to allow for an effective heat transfer into the MgO substrate. The results allow for a direct determination of the depairing current and are in good accordance with the theoretically predicted values. In intense currents the critical temperature is not only shifted to lower temperatures, as seen in conventional measurements, but the shape of the transition changes significantly. The latter effect is attributed to a reduction of the fluctuation paraconductivity with increasing electric field and is compared with our recent theoretical work. Other effects that emerge when supercritical currents are applied to high-temperature superconductors are reviewed, too.