Direct numerical simulations have been used for more than a decade to study various reduced models of plasma dynamics. This paper is devoted to a summary of the contribution made by direct numerical simulations of fluid models to the present knowledge of the basic mechanisms of anomalous transport, which have been and still largely remain the main objective of the majority of the investigations. In the first part of the work, the seemingly prevailing claim that numerical results are consistent with quasilinear theory is critically analyzed employing general arguments of turbulence theory. Then the emphasis will be shifted to some emerging physics issues in plasma turbulence which will presumably play a major role in the future numerical investigation. In particular, the problem of the determination of the correlation length, the role of the coherent structures and the question of the subcritical transition to turbulence are addressed. Finally, the subtle role of dissipation is discussed: with the support of recent numerical results and exploiting the analogy of some plasma turbulence models with those employed for ordinary fluids, it is argued that dissipation is likely to enter in a non-trivial way in the global scaling laws for plasma transport.