This is the prevailing method used in random number generators. These are generated by algorithms tuned to optimize the sequences uniform distribution , which can improve the accuracy of Monte-Carlo integration. These numbers are not independent and thus cannot be used generally. Other than uniform distributions can be generated by suitable transformations of the basic uniformly distributed sequence. Numerical libraries often offer a rich set of distributions.
All generators can initialize the sequence by a starting seed.
Storing and reloading a generator's internal state is also useful. Vector and parallel computers need special RNG methods. Which of these aspects is most important depends on the actual application, of course. What follows is a collection of material on pseudo random number generators.