Evolution of the Turbulent Interstellar Medium in Star Forming Galaxies The evolution of galaxies is driven by the continuous formation of stars out of interstellar matter (ISM). The ISM is a highly compressible and magnetized medium, characterized by Reynolds numbers, exceeding those in most terrestrial laboratories by orders of magnitude. Supernova (SN) explosions and stellar winds of massive stars are the dominant sources of kinetic energy in the ISM. Thus shear flows generate a high level of vorticity, leading to a very turbulent medium. The inherent complexity and nonlinearity of the system require a careful approach, especially for advanced numerical 3D parallel computer simulations. We have built a bottom-up model of a turbulent SN driven ISM over the years, showing (amongst others) by large scale hydro- and magnetohydrodynamical simulations (i) the existence of a large scale galactic fountain, (ii) a reduced volume filling factor of hot gas, and (iii) a considerable fraction of the gas mass residing in classical thermally unstable phases. In addition, we have recently calculated the fully time-dependent ionization structure for the 10 most abundant elements along with the hydrodynamics. Since the plasma is generally out of ionization equilibrium, the cooling function is varying both in space and time, depending on the thermal history of the plasma. The simulations are compared to some recent observations, and prospects of current and future work are discussed.