doi:10.1017/S09624929XXXXXX Printed in the United Kingdom Acta Numerica: High dimensional integration – the Quasi-Monte Carlo way

This paper is a contemporary review of QMC (“Quasi-Monte Carlo”) meth-ods, i.e., equal-weight rules for the approximate evaluation of high dimensional integrals over the unit cube [0, 1]s, where s may be large, or even infinite. Af-ter a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related themes. Among those recent developments are methods of construction of both lattices and digital nets, to yield QMC rules that have a prescribed rate of convergence for sufficiently smooth func-tions, and ideally also guaranteed slow growth (or no growth) of the worst case error as s increases. A crucial role is played by parameters called “weights”, since a careful use of the weight parameters is needed to ensure that the worst case errors in an appropriately weighted function space are bounded, or grow only slowly, as the dimension s increases. Important tools for the analysis are weighted function spaces, reproducing kernel Hilbert spaces, and discrepancy