Two codes have been developed and implemented for use on massively parallelsuper computers to simulate two-dimensional and quasigeostrophic turbulence.The codes have been found to scale well with increasing resolution and width ofthe simulations. This has allowed for the highest resolution simulations of two-dimensional and quasigeostrophic turbulence so far reported in the literature.The direct numerical simulations have focused on the statistical characteristicsof turbulent cascades of energy and enstrophy, the role of coherent vorticesand departures from universal scaling laws, theoretized more than 40 yearsago. In particular, the investigations have concerned the enstrophy and energycascade in forced and decaying two-dimensional turbulence. Furthermore, theapplicability of Charney’s hypotheses on quasigeostrophic turbulence has beentested. The results have shed light on the flow evolution at very large Reynoldsnumbers. The most important results are the robustness of the enstrophycascade in forced and decaying two-dimensional turbulence, the unexpecteddependency on an infrared Reynolds number in the spectral scaling of theenergy spectrum in the inverse energy cascade, and the validation of Charney’spredictions on the dynamics of quasigeostrophic turbulence. It has also beenshown that the scaling of the energy spectrum in the enstrophy cascade isinsensitive to intermittency in higher order statistics, but that corrections mightapply to the ”universal” Batchelor-Kraichnan constant.