Several widely used uniform random number generators have been extensively subjected to three commonly used statistical tests of uniformity and randomness. The object was 1 to examine the power of these statistical tests to discriminate between good and bad random number generators, 2 to correlate these results with recently proposed mathematical characterizations of random number generators which might also be useful in such a discrimination, and 3 to examine the effect of shuffling on the random number generators. Briefly the results show that the commonly used runs test has virtually no power to discriminate between good and bad generators, while serial tests perform better. Also shuffling does help, although much more needs to be done in this area. And finally, there is some utility to the mathematical characterizations, but many unanswered questions.