Europe has been affected by record-breaking heat waves in recent decades. Using station data and a gridded reanalysis as input, four commonly used heat wave indices, the heat wave magnitude index daily (HWMId), excess heat factor (EHF), wet-bulb globe temperature (WBGT) and universal thermal climate index (UTCI), are computed. The extremeness of historical European heat waves between 1979 and 2019 using the four indices and different metrics is ranked. A normalisation to enable the comparison between the four indices is introduced. Additionally, a method to quantify the influence of the input parameters on heat wave magnitude is introduced. The spatio-temporal behaviour of heat waves is assessed by spatial–temporal tracking. The areal extent, large-scale intensity and duration are visualized using bubble plots. As expected, temperature explains the largest variance in all indices, but humidity is nearly as important in WBGT and wind speed plays a substantial role in UTCI. While the 2010 Russian heat wave is by far the most extreme event in duration and intensity in all normalized indices, the 2018 heat wave was comparable in size for EHF, WBGT and UTCI. ... mehrInterestingly, the well-known 2003 central European heat wave was only the fifth and tenth strongest in cumulative intensity in WBGT and UTCI, respectively. The June and July 2019 heat waves were very intense, but short-lived, thus not belonging to the top heat waves in Europe when duration and areal extent are taken into account. Overall, the proposed normalized indices and the multi-metric assessment of large-scale heat waves allow for a more robust description of their extremeness and will be helpful to assess heat waves worldwide and in climate projections.