We present the first to date high resolution calculations of the ISM down to scales of 0.625 pc of the global and local ISM .
The simulations show the morphology and structure of the different ISM phases and reproduce many of the features that have been observed in the Milky Way and other galaxies .
In particular , they show that the hot gas has a moderately low volume filling factor ( \sim 20 % ) even in the absence of magnetic fields .
Also , cold gas is mainly concentrated in filamentary structures running perpendicular to the midplane forming and dissipating within \sim 10 - 12 Myr .
Compression is the dominant process for their formation , but thermal instability also plays a rôle .
Also the evolution of the Local Bubble is simulated by multi-supernova explosions ; calculated extensions after \sim 13 Myr match observations .