[en] From soft X ray measurements evidence emerges that a large volume of the local interstellar medium is filled with hot gas. The sky has now been almost entirely scanned in the soft X ray bands. Although the very contribution of a halo or/and large scale galactic components is still debated, it appears that an important part of this soft X ray emission has a local origin. Its thermal nature was strongly supported after X ray lines of some highly stripped ions were observed. This local interstellar medium is likely to have been heated by the energy released in Supernova(e) explosion(s). We review such models and discuss whether they may account for both the observed fluxes in soft X ray bands and the X ray spectral data. Special emphasis is put on the intricacy of the data, which presently makes hopeless a detailed understanding of the hot local interstellar medium. Consistency of such models with key parameters (such as OVI column density) is also studied