Wavelet theory arose at the intersection of pure mathematics, applied mathematics and digital signal processing. Wavelet systems have several advantages compared with other systems used as approximation tools. One of them is the so-called time-frequency localization property: wavelet functions and their Fourier transformations rapidly decay at infinity. This property allows to analyze and compress data effectively. Also, there exist fast algorithms for wavelet decomposition. Review of the problems, connected to the construction of wavelet systems with desirable properties (such as compact support, symmetry, smoothness) is presented in the report. The general method for the construction of such systems in the multivariate case is given.