This colloquium is inspired by the legacy of Steve A. Orszag, for both themes, statistical theory and pseudo-spectral DNS. The former ranges from Spectral Linear Theory to triadic nonlinear closures, with wave turbulence theory and generalised EDQNM (Eddy Damped Quasi-Normal Markovian) closure as examples. The latter is restricted to new challenges
in fundamental aspects to turbulence, permitted by pseudo-spectral DNS techniques but connected with theory and modelling, with e.g. applications to subgrid scales, for Large Eddy Simulation, and to very large scales, in the infrared limit. Turbulence is also considered interacting with rotation, shear, density-stratication, magnetohydrodynamics.