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 stochasticmodels.- 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.