Much of the recent work dealing with quasi-random methods has been aim
ed at establishing the best possible asymptotic rates of convergence t
o zero of the error resulting when a finite-dimensional integral is re
placed by a finite sum of intergrand values. In contrast with this per
spective to concentrate on asymptotic convergence rates, this paper em
phasizes quasi-random methods that are effective for all sample sizes.
Throughout the paper, the problem of estimating finite-dimensional in
tegrals is used to illustrate the major ideas, although much of what i
s done applies equally to the problem of solving certain Fredholm inte
gral equations. Some new techniques, based on error-reducing transform
ations of the integrand, are described that have been shown to be usef
ul both in estimating high-dimensional integrals and in solving integr
al equations. These techniques illustrate the utility of carrying over
to the quasi-Monte Carlo method certain devices that have proven to b
e very valuable in statistical (pseudorandom) Monte Carlo applications
.