The existence of soft excess emission originating from clusters of galaxies, defined as emission detected below 1 keV as an excess over the usual thermal emission from hot intracluster gas (hereafter the ICM) has been claimed since 1996. Soft excesses are particularly important to detect because they may (at least partly) be due to thermal emission from the Warm-Hot Intergalactic Medium, where as much as half of the baryons of the Universe could be. They are therefore of fundamental cosmological importance.
Soft excess emission has been observed (and has also given rise to controversy) in a number of clusters, mainly raising the following questions: 1) Do clusters really show a soft excess? 2) If so, from what spatial region(s) of the cluster does the soft excess originate? 3) Is this excess emission thermal, originating from warm-hot intergalactic gas (at temperatures of ~ 106 K), or non-thermal, in which case several emission mechanisms have been proposed. Interestingly, some of the non-thermal mechanisms suggested to account for soft excess emission can also explain the hard X-ray emission detected in some clusters, for example by RXTE and BeppoSAX (also see Petrosian et al. 2008 - Chapter 10, this volume; Rephaeli et al. 2008 - Chapter 5, this volume).
Several instruments have been used to search for soft excess emission: EUVE, ROSAT and BeppoSAX in the 1990's, and presently XMM-Newton, Chandra and Suzaku. We will briefly present a history of these detections, emphasizing the difficulties to extract such weak signal and the underlying hypotheses, and summarising what is known on each of the observed clusters. For clarity, results will be presented separately for the various satellites. Finally we summarise and discuss the various models proposed to account for Extreme Ultraviolet (hereafter EUV) emission with their pros and cons.