Wavelet analysis is a new method called ‘numerical microscope’ in signal and image processing. It has the desirable advantages of multi-resolution properties and various basis functions, which fulfill an enormous potential for solving partial differential equations (PDEs). The numerical analysis with wavelet received its first attention in 1992, since then researchers have shown growing interest in it. Various methods including wavelet weighted residual method (WWRM), wavelet finite element method (WFEM), wavelet boundary method (WBM), wavelet meshless method (WMM) and wavelet-optimized finite difference method (WOFD), etc. have acquired an important role in recent years. This paper aims to make a comprehensive review and classification on wavelet-based numerical analysis and to note their merits, drawbacks, and future directions. And thus the present review helps readers identify research starting points in wavelet-based numerical analysis and guides researchers and practitioners. •Wavelet analysis is a method called numerical microscope in signal processing and numerical analysis.•A comprehensive review on wavelet-based numerical analysis is made and their merits, drawbacks, and future directions are noted.•From the aspect of algorithm construction, the wavelet-based numerical analysis methods are categorized into five types.