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preface chapter 1 introduction 1.1 turbulence and symmetries 1.2 outline of the book chapter 2 symmetries and conservation laws 2.1 periodic boundary conditions 2.2 symmetries 2.3 conservation laws 2.4 energy budget scale-by-scale chapter 3 why a probabilistic description of turbulence? 3.1 there is something predictable in a turbulent signal 3.2 a model for deterministic chaos 3.3 dynamical systems 3.4 the navier-stokes equation as a dynamical system chapter 4 probabilistic tools: a survey 4.1 random variables 4.2 random functions 4.3 statistical symmetries 4.4 ergodic results 4.5 the spectrum of stationary random functions chapter 5 two experimental laws of fully developed turbulence 5.1 the two-thirds law 5.2 the energy dissipation law chapter 6 the kolmogorov 1941 theory 6.1 kolmogorov 1941 and symmetries 6.2 kolmogorov's four-fifths law 6.2.1 the k/trm/m-howarth-monin relation for anisotropic turbulence 6.2.2 the energy flux for homogeneous turbulence 6.2.3 the energy flux for homogeneous isotropic turbulence 6.2.4 from the energy flux relation to the four-fifths law 6.2.5 remarks on kolmogorov's four-fifths law 6.3 main results of the kolmogorov 1941 theory 6.3.1 the kolmogorov-obukhov law and the structure func-tions 6.3.2 effect of a finite viscosity: the dissipation range 6.4 kolmogorov and landau: the !ack of universality 6.4.1 the original formulation of landau's objection 6.4.2 a modern reformulation of landau's objection 6.4.3 kolmogorov and landau reconciled? 6.5 historical remarks on the kolmogorov 1941 theory chapter 7 phenomenology of turbulence in the sense of kolmogorov 1941 7.1 introduction 7.2 basic tools of phenomenology 7.3 the richardson cascade and the !ocalness of interactions 7.4 reynolds numbers and degrees of freedom 7.5 microscopic and macroscopic degrees of freedom 7.6 the distribution of velocity gradients 7.7 the law of decay of the energy 7.8 beyond phenomenology: finite-time blow-up of ideal flow chapter 8 intermittency 8.1 introduction 8.2 self-similar and intermittent random functions 8.3 experimental results on intermittency 8.4 exact results on intermlttency 8.5 intermittency models based on the velocity 8.5.1 the¦Â-model 8.5.2 the bifractal model 8.5.3 the multifractal model 8.5.4 a probabilistic reformulation of the multifractal model 8.5.5 the intermediate dissipation range and multifractal universality 8.5.6 the skewness and the flatness of velocity derivatives according to the multifractal model 8.6 lntermittency models based on the dissipation 8.6.1 multifractal dissipation 8.6.2 bridging multifractality based on the velocity and mul-tifractality based on the dissipation 8.6.3 random cascade models 8.6.4 large deviations and multifractality 8.6.5 the lognormal model and its shortcomings 8.7 shell models 8.8 historical remarks on fraetal intermitteney models 8.9 trends in intermitteney research 8.9.1 vortex filaments: the sinews of turbulence? 8.9.2 statistical signature of vortex filaments: dog or tail? 8.9.3 the distribution of velocity increments chapter 9 further reading: a guided tour 9.1 introduction 9.2 books on turbulence and fluid mechanics 9.3 mathematical aspects of fully developed turbulence 9.4 dynamical systems, fractals and turbulence 9.5 closure, functional and diagrammatic methods 9.5.1 the hopfequation 9.5.2 functional and diagrammatic methods 9.5.3 the direct interaction approximation 9.5.4 closures and their shortcomings 9.6 eddy viscosity, multiscale methods and renormalization 9.6.1 eddy viscosity: a very old idea 9.6.2 multiscale methods 9.6.3 applications of multiscale methods in turbulence 9.6.4 renormalization group (rg) methods 9.7 two-dimensional turbulence 9.7.1 cascades and vortices 9.7.2 two-dimensional turbulence and statistical mechanics. 9.7.3 conservative dynamics 'punctuated' by dissipative events 9.7.4 from flatland to three-dimensional turbulence references author index subject index