This is a critical exposition and analysis of the modern developments in the theory of turbulent motion of an incompressible fluid. We begin with the review of the mixture length theories based upon Reynolds' equations of mean motion. Secondly, we analyze the principal contributions to the theory of homogeneous isotropic turbulence. Thirdly, we discuss the treatment of the general turbulent shear flow by means of Reynolds' equations of mean motion and the dynamical equations of velocity correlations which are derived from the equations of turbulent velocity fluctuation. We also point out at the same time that although this method yields theoretical results which are in better agreement with experiment than the results of the mixture length theories and furthermore the theory also leads to the theoretical distributions ot the mean squares of velocity fluctuation, on account of the presence of the higher order velocity correlations in the equations, it continuously leads to unclosed systems of differential equations and hence meets difficulties which are difficult to overcome. Therefore, based upon the above retrospect of the developments of the theory of turbulence and the recent work on the vorticity structure of the homogeneous isotropic turbulence in its final period of decay, we finally propose a new approach to the turbulence problem: The basic component motion of turbulence is vortex motion due to the action of viscosity of the fluid. The dynamical equations which govern the vortex motion of turbulence are Reynolds' equations of mean motion and the equations of velocity fluctuation derived from the Navier-Stokes equations by the averaging process. We also emphasize the importance of Reynolds' recognition that the turbulent motion of a fluid can be separated into the mean motion and fluctuation. The future theoretical investigation is to look for the vortex motions which are solutions of these two sets of equations. In order to make the solutions of the problem unique and comparable with experimental measurements, they should also satisfy statistical conditions on the distribution of vortices analogous to Kol-mogoroffs condition in his statistical theory of locally isotropic turbulence at high Reynolds number turbulent flows.