I discuss the role of self-gravity and radiative heating and cooling in shaping the nature of the turbulence in the interstellar medium (ISM) of our galaxy. The heating and cooling cause it to be highly compressible, and, in some regimes of density and temperature, to become thermally unstable, tending to spontaneously segregate into warm/diffuse and cold/dense phases. On the other hand, turbulence is an inherently mixing process, tending to replenish the density and temperature ranges that would be forbidden under thermal processes alone. The turbulence in the ionized ISM appears to be transonic (i.e, with Mach numbers $Ms sim 1$), and thus to behave essentially incompressibly. However, in the neutral medium, thermal instability causes the sound speed of the gas to fluctuate by up to factors of $sim 30$, and thus the flow can be highly supersonic with respect to the dense/cold gas, although numerical simulations suggest that this behavior corresponds more to the ensemble of cold clumps than to the clumps internal velocity dispersion. Finally, coherent large-scale compressions in the warm neutral medium (induced by, say, the passage of spiral arms or by supernova shock waves) can produce large, dense molecular clouds that are subject to their own self-gravity, and begin to contract gravitationally. Because they are populated by nonlinear density fluctuations, whose local free-fall times are significantly smaller than that of the whole cloud, the fluctuations terminate their collapse earlier, giving rise to a regime of hierarchical gravitational fragmentation, with small-scale collapses occurring within larger-scale ones. Thus, the turbulence in molecular clouds may be dominated by a gravitationally contracting component at all scales.
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