High-dimensional problems are ubiquitous in such diverse fields as quantum mechanics, plasma physics, uncertainty quantification, or radiation therapy. The numerical solution of such problems is extremely costly due to the unfavorable scaling of the number of degrees of freedom with dimension; the so-called cure of dimensionality. In order to alleviate this, a number of complexity reduction techniques (such as Monte Carlo methods, sparse grids, low-rank approximation, etc.) have been developed. In this minisymposium we will report on recent advances of such methods and their applications.