The problem is discussed of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure. This problem is investigated for elements of finite-dimensional spaces. Also, discretization of the uniform norm of functions in a given finite-dimensional subspace of continuous functions is studied. Special attention is given to the case of multivariate trigonometric polynomials with frequencies (harmonics) in a finite set with fixed cardinality. Both new results and a survey of known results are presented. Bibliography: 47 titles. © 2019 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.