Monte Carlo simulation is one of the main applications involving the u
se of random number generators. It is also one of the best methods of
testing the randomness properties of such generators, by comparing res
ults of simulations using different generators with each other, or wit
h analytic results. Here we compare the performance of some popular ra
ndom number generators by high precision Monte Carlo simulation of the
2-d Ising model, for which exact results are known, using the Metropo
lis, Swendsen-Wang, and Wolff Monte Carlo algorithms. Many widely used
generators that perform well in standard statistical tests are shown
to fail these Monte Carlo tests.