Interstellar turbulence significantly influences the structure and motion of gas across various temperature and density regimes. This review summarizes observations, theory, and simulations of interstellar turbulence and their implications for astrophysics. The first part discusses diagnostics for turbulence in the cool interstellar medium, energy sources, and turbulence theory. Energy sources include supernovae, superbubbles, and smaller-scale processes like gravitational instabilities and cosmic ray streaming. Turbulence theory covers fluid equations, solenoidal and compressible modes, scaling arguments, and the differences between incompressible and compressible turbulence. Numerical simulations have reproduced observed scaling relations and predicted fast decay rates for supersonic MHD turbulence. Thermal instabilities and phases have new interpretations in supersonically turbulent media. Large-scale models with self-gravity, magnetic fields, and star formation begin to resemble the observed interstellar medium. Self-gravity plays a key role in turbulent gas evolution, influencing star cluster formation, the stellar mass function, and the effects of magnetic fields. The review highlights progress in understanding the interstellar medium and lists outstanding problems. Observations and simulations show power-law scaling in turbulence, with slopes similar to Kolmogorov scaling. Turbulence diagnostics include structure functions, power spectra, and delta variance. These methods reveal complex structures and scaling in the interstellar medium. Power sources for turbulence include stellar winds, supernovae, galactic rotation, and gravitational instabilities. Theoretical models of turbulence consider incompressible and compressible cases, with energy cascades and dissipation. The review emphasizes the complexity of turbulence in the interstellar medium and the need for further study.Interstellar turbulence significantly influences the structure and motion of gas across various temperature and density regimes. This review summarizes observations, theory, and simulations of interstellar turbulence and their implications for astrophysics. The first part discusses diagnostics for turbulence in the cool interstellar medium, energy sources, and turbulence theory. Energy sources include supernovae, superbubbles, and smaller-scale processes like gravitational instabilities and cosmic ray streaming. Turbulence theory covers fluid equations, solenoidal and compressible modes, scaling arguments, and the differences between incompressible and compressible turbulence. Numerical simulations have reproduced observed scaling relations and predicted fast decay rates for supersonic MHD turbulence. Thermal instabilities and phases have new interpretations in supersonically turbulent media. Large-scale models with self-gravity, magnetic fields, and star formation begin to resemble the observed interstellar medium. Self-gravity plays a key role in turbulent gas evolution, influencing star cluster formation, the stellar mass function, and the effects of magnetic fields. The review highlights progress in understanding the interstellar medium and lists outstanding problems. Observations and simulations show power-law scaling in turbulence, with slopes similar to Kolmogorov scaling. Turbulence diagnostics include structure functions, power spectra, and delta variance. These methods reveal complex structures and scaling in the interstellar medium. Power sources for turbulence include stellar winds, supernovae, galactic rotation, and gravitational instabilities. Theoretical models of turbulence consider incompressible and compressible cases, with energy cascades and dissipation. The review emphasizes the complexity of turbulence in the interstellar medium and the need for further study.