In an era of frequent extreme weather and global warming, obtaining precise, fine-grained near-surface weather forecasts is increasingly essential for human activities. Downscaling (DS), a crucial task in meteorological forecasting, enables the reconstruction of high-resolution meteorological states for target regions from global-scale forecast results. Previous downscaling methods, inspired by CNN and Transformer-based super-resolution models, lacked tailored designs for meteorology and encountered structural limitations. Notably, they failed to efficiently integrate topography, a crucial prior in the downscaling process. In this paper, we address these limitations by pioneering the selective state space model into the meteorological field downscaling and propose a novel model called MambaDS. This model enhances the utilization of multivariable correlations and topography information, unique challenges in the downscaling process while retaining the advantages of Mamba in long-range dependency modeling and linear computational complexity. Through extensive experiments in both China mainland and the continental United States (CONUS), we validated that our proposed MambaDS achieves state-of-the-art results in three different types of meteorological field downscaling settings. We will release the code subsequently.