Mersenne Twister was maybe the first in a class of random number generators that has lots of state -- an order of magnitude more than previous designs. Each time a number is pulled from it, some of the state is stirred, and the next number comes from mostly other bits, and stirs others. They have to be fast, so you can't touch too many bits per number extracted; taking about the same time for each number is nice, too.
So, one measure of generators is how many numbers you pull before you get the same sequence again. MT's cycle is very long, so in practice you never see a repeat, even if you see the same number many times. (In many simpler generators, seeing 3 then 8 means next time you see 3, the next number will be 8. A great deal of simulation was done with such generators.) The numbers from an MT satisfy many different measures of apparent randomness.
Monte Carlo investigations consume very many numbers. They might use the numbers in a more or less periodic way, so that any match-up between cycles in the problem and cycles in the generator can skew the results. The main MT cycle is very long, so any skewed results probably point to lesser cycles as the bits stirred are later encountered again. But it's hard to imagine a way to detect such cycles deliberately from the bits you get out. Encountering a process that finds them accidentally is amazing.
