Abstract: Data assimilation, as a key scientific issue in accurately depicting the state of the atmosphere, oceans, and climate, was proposed over a hundred years ago. However, it is the rapid advancement of computer technology in recent decades that has enabled the implementation and application of some cutting-edge methods and techniques. Data assimilation is an interdisciplinary applied science closely related to meteorology, physics, mathematics, and computer science, and its development is closely linked to the progress of these disciplines. It would be of great interest to compare advanced methods and theories of data assimilation with achievements of these disciplines, hoping to summarize past progress and provide guidance for future development. This article, from the perspective of applied mathematics, delves into the essence of mainstream assimilation algorithms, clarifies common misconceptions, and proposes key directions for future research.