This paper examines the evolution of closing the Reynolds-averaged Navier-Stokes equations by approximating the Reynolds stresses via the second-moment transport equations themselves. This strategy first proposed by Rotta is markedly in contrast to the more usual approach of computing an effective “turbulent viscosity” to deduce the turbulent stresses as in a Newtonian fluid in laminar motion. This paper covers the main elements in the development of this approach and shows examples of applications in complex shear flows that collectively include the effects of three-dimensional straining, force fields, and time dependence that affect the flow evolution in ways that cannot be readily mimicked with an eddy viscosity model.