As far as I'm aware, most implementations of pseudorandom number generation in practice use methods such as linear shift feedback registers (LSFRs), or these "Mersenne Twister" algorithms. While they pass lots of (heuristic) statistical tests, there are no theoretical guarantees that they look pseudorandom to, say, all efficiently computable statistical tests. Yet these methods are used indiscriminately in all sorts of applications, ranging from cryptographic protocols to scientific computing to banking (probably). I find it somewhat worrisome that we have little to no guarantee about whether these applications work as intended (because any sort of analysis would likely have assumed true randomness as input).
On the other hand, complexity theory and cryptography provide a very rich theory of pseudorandomness, and we even have candidate constructions of pseudorandom generators that would fool ANY efficient statistical test you could come up with, using candidate one way functions.
My question is: has this theory made its way into practice? I would hope that for important uses of randomness, such as cryptography or scientific computing, theoretically sound PRGs are used.
As an aside, I could find some limited analysis of how well popular algorithms such as quicksort work when using LSFRs as a source of randomness, and apparently they work well. See Karloff and Raghavan's "Randomized algorithms and pseudorandom numbers".