Lie group symmetry constitutes a powerful modelling tool in many scientific areas. They allow, for instance, the computation of Green function of linear equations. Through Noether's theorem, it is also known that each symmetry of an equation corresponds to a conservation law. In addition, symmetries are extensively used in litterature to compute self-similar solutions of various equations. In turbulence, vortex solutions of the Navier-Stokes equations was found as special self-similar solutions. Finally, we mention that the symmetries may give an information on the large-time behaviour of the solution. To some extent, the symmetries traduces the physics of the equations.

In this presentation, other applications of Lie symmetry group theory to the modelling and simulation of turbulent flows are shown. More precisely, the Lie symmetries of Reynolds Averaged Navier-Stokes equations are used to retrieve classical but also to exhibit new wall and scaling laws of non-isothermal flows. Next, the development of symmetry preserving closure models for large-eddy flow simulation methods is presented. At last, we show that, at discrete scale, symmetry preservation leads to more robust numerical scheme. To this aim, we examine the construction of invariantized discretization schemes with Cartan's moving frame method.