Data assimilation is a vital component in modern global medium-range weather forecasting systems to obtain the best estimation of the atmospheric state by combining the short-term forecast and observations. Recently, AI-based data assimilation approaches have attracted increasing attention for their significant advantages over traditional techniques in terms of computational consumption. However, existing AI-based data assimilation methods can only handle observations with a specific resolution, lacking the compatibility and generalization ability to assimilate observations with other resolutions. Considering that complex real-world observations often have different resolutions, we propose the Fourier Neural Processes (FNP) for arbitrary-resolution data assimilation in this paper. Leveraging the efficiency of the designed modules and flexible structure of neural processes, FNP achieves state-of-the-art results in assimilating observations with varying resolutions, and also exhibits increasing advantages over the counterparts as the resolution and the amount of observations increase. Moreover, our FNP trained on a fixed resolution can directly handle the assimilation of observations with out-of-distribution resolutions and the observational information reconstruction task without additional fine-tuning, demonstrating its excellent generalization ability across data resolutions as well as across tasks. Code is available at https://github.com/OpenEarthLab/FNP.