Quasi-Monte Carlo (QMC) has been shown to be superior to Monte Carlo (MC) for financial computations. This may be due to the non-isotropic nature of high-dimensional integrals arising in finance.
Recently we tested an isotropic problem from physics and found that again QMC was far superior to MC. See Faster Evaluation of Multidimensional Integrals, Computers in Physics, November, 1997, 574-578 (with A. Papageorgiou).
An open question is to characterize for which classes of integrals QMC is superior to MC.
The expected error of MC for computing multivariate integrals is proportional to n-1/2 if the integrand is evaluated at n randomly chosen points. What is the expected error if pseudo-random points are used? An analysis may be found in The Monte Carlo Algorithm with a Pseudo-random Generator , Mathematics of Computation, Vol. 58, 1992, 303-339 (with H. Wozniakowski).