The traditional Fourier analysis and continuous wavelet method will produce relatively enormous errors due to the interference of Gibbs phenomenon. To solve this problem, Qi Dongxu proposed the research topic of discontinuous orthogonal function systems, among which U-system and V-system are two typical discontinuous complete orthogonal function systems. In terms of the mathematical theory, U-system and V-system are the results of the extension of the well-known Walsh function and Haar function from piecewise constant to piecewise k degree polynomial, respectively. The most important feature of U-system is that there are both smooth functions and discontinuous functions at various levels in the function system. Therefore, U- and V- systems can process both continuous and discontinuous information, making up for the shortcomings of Fourier analysis and continuous wavelet to a certain extent. This paper reviewed U- and V- systems from two aspects: theory and application. Theoretically, firstly, the construction methods of univariate U-system and V-system were introduced, respectively, then the construction methods of V-system on triangular domain were introduced, and finally the main properties of U- and V-systems were introduced. In terms of application, some representative cases of applications were introduced.