The wide majority of statistical tools used in turbulence to date are often ill-adapted to describe strongly anisotropic turbulent flows submitted to body forces or mean gradients. Anisotropy and inhomogeneity are generally assumed at large scale only, in contradiction with recent results in rapidly rotating flows. The anisotropic morphology of turbulence is intimately linked to dimensionality, and different dimensionalities yield different dynamical aspects. For flows with a dominant direction, called here the axial direction, whose statistical properties are axisymmetric, these various and multifold morphological and dynamical aspects are collected under the misleading vocable 'quasi-two-dimensional'. For instance, rotating homogeneous turbulence can be shown to decay freely to an asymptotic state differing from two-dimensional turbulence. A general description of strongly anisotropic flows has been developed in the LMFA, with a spectral formalism that includes a minimal number of components, e.g. based on the poloidal-toroidal splitting of the divergence-free velocity field. The corresponding energy spectra depend on two wavenumber components. An equivalent decomposition and description can be made in physical space, a part of the present proposal. The interscale energy transfer itself reflects an anisotropic dynamics via Lin type equations written in terms of triadic exchanges. The corresponding third-order interaction coefficients and nonlinear spectral energy transfer potentially carry more information than third-order structure functions in physical space. In addition, spectral formalism can easily remove pressure fluctuations from the set of unknowns. An important goal of this project is to use the most general axisymmetric description of turbulent fields in Fourier space for deriving more global quantities: classical two-dimensional spectra, second- and third-order anisotropic structure functions, directional length scales etc. In turn, statistical quantities in physical space, obtained from experiments, can afford to include inhomogeneous effects mainly induced by confinement, and help us to delineate the limits of strongly anisotropic homogeneous theory. This project will therefore blend together results from numerical simulations, from a selected set of experiments, and from statistical modelling, for completeness. Numerical simulations are necessary to reach any kind of statistics of the turbulent field, some of which are not amenable to experimental measurements, although the latest PIV and acoustic probing techniques will be used here. Theoretical developments based on eigenmodes decomposition, and the statistical models that rely on them, which we will develop, permit a more phenomenological analysis. We will apply these methods to investigate the statistical properties of flows of different natures: rotating turbulence with or without walls, channel flow with wall normal or spanwise rotation, the round jet and recirculating flows, all of which provide a different test-case for the validation of our theoretical description of axisymmetric turbulence.