Turbulent flow arises in a wide variety of natural and technological situations. While the full richness of turbulence is appreciated qualitatively, a quantitatively accurate prediction is often outside the scope of direct numerical computations. As an alternative, filtered flow descriptions, such as large-eddy simulation (LES), have been proposed and studied intensively, promising a combination of accuracy and computational feasibility. Many heuristic closure models for small-scale turbulence have been put forward to represent their dynamic effects on the large-scale characteristics of the flow. While these models are often effective in reducing the dynamic complexity of the LES approach, accuracy limitations of LES are a matter of ongoing discussion. In this presentation, the alternative offered by mathematical regularization, pioneered already by Leray in the 1930s, is explored. Following the regularization approach for the nonlinear convective terms, the closure model is uniquely connected to the underlying regularization principle, thereby by-passing the heuristic closure modeling that is characteristic of the filtering approach to LES. A number of regularization models will be reviewed and their performance in homogeneous isotropic turbulence and in turbulent mixing will be discussed with particular emphasis to flow at high Reynolds numbers. It will be shown that regularization methods that account for effective dissipation can be accurate at strongly reduced computational costs.