This is a review of pseudorandom number generators (RNG's) of the highest quality, suitable for use in the most demanding Monte Carlo calculations. All the RNG's we recommend here are based on the Kolmogorov-Anosov theory of mixing in classical mechanical systems, which guarantees under certain conditions and in certain asymptotic limits, that points on the trajectories of these systems can be used to produce random number sequences of exceptional quality. We outline this theory of mixing and establish criteria for deciding which RNG's are sufficiently good approximations to the ideal mathematical systems that guarantee highest quality. The well-known RANLUX (at highest luxury level) and its recent variant RANLUX++ are seen to meet our criteria, and some of the proposed versions of MIXMAX can be modified easily to meet the same criteria.1 High-quality random numbersWe are concerned here with pseudorandom number generators (RNG's), in particular those of the highest quality. It turns out to be difficult to find an operational definition of randomness that can be used to measure the quality of a RNG, that is the degree of independence of the numbers in a given sequence, or to prove that they are indeed independent. The situation for traditional RNG's (not based on Kolmogorov-Anasov mixing) is well described by Donald Knuth in [2]. The book contains a wealth of information about random number generation, but nothing about where the randomness comes from, or how to measure the quality (randomness) of a generator. Now with hindsight, it is not surprising that all the widely-used generators described there were later found to have defects (failing tests of randomness and/or giving incorrect results in Monte Carlo (MC) calculations), with the notable exception of RANLUX, which Knuth does mention briefly in the third edition, but without describing the new theoretical basis.