This textbook presents a modern account of turbulence, one of the greatest challenges in physics. The state-of-the-art is put into historical perspective five centuries after the first studies of Leonardo and half a century after the first attempt by A. N. Kolmogorov to predict the properties of flow at very high Reynolds numbers. Such ?fully developed turbulence? is ubiquitous in both cosmical and natural environments, in engineering applications and in everyday life. The intended readership for the book ranges from first-year graduate students in mathematics, physics, astrophysics, geosciences and engineering, to professional scientists and engineers. Elementary presentations of dynamical systems ideas, of probabilistic methods (including the theory of large deviations) and of fractal geometry make this a self-contained textbook.

1. Introduction
 2. Symmetries and conservation laws
 3. Why a probabilistic description of turbulence?
 4. Probabilistic tools: a survey
 5. Two experimental laws of fully developed turbulence
 6. The Kolmogorov 1941 theory
 7. Kolmogorov and Landau: the lack of universality
 8. Phenomenology of turbulence in the sense of Kolmogorov 1941
 9. Intermittency
 10. Further reading: a guided tour
 Bibliography.