An overview is presented of the main properties of the interstellar medium. Evidence is summarized that the interstellar medium is highly turbulent, driven on different length scales by various energetic processes. Large-scale turbulence determines the formation of structures like filaments and shells in the diffuse interstellar medium. It also regulates the formation of dense, cold molecular clouds. Molecular clouds are now believed to be transient objects that form on timescales of order 1e7 yrs in regions where HI gas is compressed and cools. Supersonic turbulence in the compressed HI slab is generated by a combination of hydrodynamical instabilities, coupled with cooling. Turbulent dissipation is compensated by the kinetic energy input of the inflow. Molecular hydrogen eventually forms when the surface density in the slab reaches a threshold value of 1e21 cm^-2 at which point further cooling triggers the onset of star formation by gravitational collapse. A few Myrs later, the newly formed stars and resulting supernovae will disperse their molecular surrounding and generate new expanding shells that drive again turbulence in the diffuse gas and trigger the formation of a next generation of cold clouds. Although a consistent scenario of interstellar medium dynamics and star formation is emerging many details are still unclear and require more detailed work on microphysical processes as well as a better understanding of supersonic, compressible turbulence.

13 pages, 4 figures, to appear in "Statistical Mechanics of Non-Extensive Systems", eds. F. Combes and R. Robert (Elsevier)