The chapter discusses the renormalization group (RNG) analysis of turbulence, focusing on the application of RNG techniques to hydrodynamic turbulence. The primary objective is to understand and extend a recent theory based on dynamic RNG methods, particularly the work by Yakhnot and Orszag (1986). This theory aims to calculate eddy viscosity and predict flow constants such as the Kolmogorov constant without adjustable parameters. The RNG transformation involves two steps: averaging to eliminate small scales and rescaling space. A set of equations is renormalizable if it remains unchanged by the RNG transformation, implying scale invariance. The method is iterative, and the physics at fixed points describes the behavior of more general cases. The basic premise of the RNG analysis of turbulence is the equivalence between inertial range solutions of the Navier-Stokes equations and homogeneous isotropic flow driven by a Gaussian random force. The analysis yields scaling laws for velocity correlations and the energy spectrum, recovering the Kolmogorov spectrum when the exponent \( y \) is set to 3. A revised RNG analysis by Yakhot and Orszag shows that scale elimination alone can find both scaling laws and amplitudes, leading to the prediction of the Kolmogorov constant. The effect of small scales on large scales is determined by an eddy viscosity, which depends on the low wavenumber cutoff \( k_c \). The chapter also evaluates universal flow constants, such as the Kolmogorov constant and the Batchelor constant, using power series expansions at the fixed point. Future goals include further analysis of the RNG model, assessing the importance of neglected terms, exploring different forcing functions, extending the theory to non-homogeneous and isotropic flows, and testing the RNG sub-grid and \(\kappa-\epsilon\) models.The chapter discusses the renormalization group (RNG) analysis of turbulence, focusing on the application of RNG techniques to hydrodynamic turbulence. The primary objective is to understand and extend a recent theory based on dynamic RNG methods, particularly the work by Yakhnot and Orszag (1986). This theory aims to calculate eddy viscosity and predict flow constants such as the Kolmogorov constant without adjustable parameters. The RNG transformation involves two steps: averaging to eliminate small scales and rescaling space. A set of equations is renormalizable if it remains unchanged by the RNG transformation, implying scale invariance. The method is iterative, and the physics at fixed points describes the behavior of more general cases. The basic premise of the RNG analysis of turbulence is the equivalence between inertial range solutions of the Navier-Stokes equations and homogeneous isotropic flow driven by a Gaussian random force. The analysis yields scaling laws for velocity correlations and the energy spectrum, recovering the Kolmogorov spectrum when the exponent \( y \) is set to 3. A revised RNG analysis by Yakhot and Orszag shows that scale elimination alone can find both scaling laws and amplitudes, leading to the prediction of the Kolmogorov constant. The effect of small scales on large scales is determined by an eddy viscosity, which depends on the low wavenumber cutoff \( k_c \). The chapter also evaluates universal flow constants, such as the Kolmogorov constant and the Batchelor constant, using power series expansions at the fixed point. Future goals include further analysis of the RNG model, assessing the importance of neglected terms, exploring different forcing functions, extending the theory to non-homogeneous and isotropic flows, and testing the RNG sub-grid and \(\kappa-\epsilon\) models.