I Introduction to turbulence in fluid mechanics -- 1 Is it possible to define turbulence? -- 2 Examples of turbulent flows -- 3 Fully developed turbulence -- 4 Fluid turbulence and "chaos" -- 5 "Deterministic" and statistical approaches -- 6 Why study isotropic turbulence? -- II Basic fluid dynamics -- 1 Eulerian notation and Lagrangian derivatives -- 2 The continuity equation -- 3 The conservation of momentum -- 4 The thermodynamic equation -- 5 The incompressibility assumption -- 6 The dynamics of vorticity -- 7 The generalized Kelvin theorem -- 8 The Boussinesq equations -- 9 Internal inertial-gravity waves -- 10 Barré de Saint-Venant equations -- III Transition to turbulence -- 1 The Reynolds number -- 2 The Rayleigh number -- 3 The Rossby number -- 4 The Froude Number -- 5 Turbulence, order and chaos -- IV The Fourier space -- 1 Fourier representation of a flow -- 2 Navier-Stokes equations in Fourier space -- 3 Boussinesq equations in the Fourier space -- 4 Craya decomposition -- 5 Complex helical waves decomposition -- V Kinematics of homogeneous turbulence -- 1 Utilization of random functions -- 2 Moments of the velocity field, homogeneity and stationarity -- 3 Isotropy -- 4 The spectral tensor of an isotropic turbulence -- 5 Energy, helicity, enstrophy and scalar spectra -- 6 Alternative expressions of the spectral tensor -- 7 Axisymmetric turbulence -- VI Phenomenological theories -- 1 The closure problem of turbulence -- 2 Karman-Howarth equations in Fourier space -- 3 Transfer and Flux -- 4 The Kolmogorov theory -- 5 The Richardson law -- 6 Characteristic scales of turbulence -- 7 The skewness factor -- 8 The internal intermittency -- VII Analytical theories and stochastic models -- 1 Introduction -- 2 The Quasi-Normal approximation -- 3 The Eddy-Damped Quasi-Normal type theories -- 4 The stochastic models -- 5 Phenomenology of the closures -- 6 Numerical resolution of the closure equations -- 7 The enstrophy divergence and energy catastrophe -- 8 The Burgers-M.R.C.M. model -- 9 Isotropic helical turbulence -- 10 The decay of kinetic energy -- 11 E.D.Q.N.M. and R.N.G. techniques -- VIII Diffusion of passive scalars -- 1 Introduction -- 2 Phenomenology of the homogeneous passive scalar diffusion -- 3 The E.D.Q.N.M. isotropic passive scalar -- 4 The decay of temperature fluctuations -- 5 Lagrangian particle pair dispersion -- IX Two-dimensional and quasi-geostrophic turbulence -- 1 Introduction -- 2 The quasi-geostrophic theory -- 3 Two-dimensional isotropic turbulence -- 4 Diffusion of a passive scalar -- 5 Geostrophic turbulence -- X Absolute equilibrium ensembles -- 1 Truncated Euler Equations -- 2 Liouville's theorem in the phase space -- 3 The application to two-dimensional turbulence -- 4 Two-dimensional turbulence over topography -- XI The statistical predictability theory -- 1 Introduction -- 2 The E.D.Q.N.M. predictability equations -- 3 Predictability of three dimensional turbulence -- 4 Predictability of two-dimensional turbulence -- XII Large-eddy simulations -- 1 The direct numerical simulation of turbulence -- 2 The Large Eddy Simulations -- 3 L.E.S. of 3-D isotropic turbulence -- 4 L.E.S. of two-dimensional turbulence -- XIII Towards "real world turbulence" -- 1 Introduction -- 2 Stably Stratified Turbulence -- 3 The Mixing Layer -- 4 Conclusion -- References.