Abstract: The Advanced Regional Eta-coordinate Model (AREM) is featured as a useful tool to simulate and forecast meso-scale systems such as torrential rainfall and typhoons in China. However, the hydrostatic approximation has curbed its further development, which is more and more noticeable with the increasingly high model resolution. In this paper, the authors present a non-hydrostatic extension to AREM through the consideration of higher-order correctness due to the vertical acceleration. The AREM non-hydrostatic dynamics employs a primitive Euler equation system of motion and effectively uses the deduction of standard stratification and IAP (Institute of Atmospheric Physics) transformation of the current hydrostatic model. Also, the non-hydrostatic version of AREM is formulated in the spherical colatitude-longitude mesh and discretized in the Arakawa-E grid and a vertical η coordinate system. The prognostic equations are split into two parts, that is, the quasi-hydrostatic system and non-hydrostatic system, which facilitates efficient integration of the dynamic core. The sound wave associated with the non-hydrostatic system is calculated through the Thomas algorithm and iteration method. This approach to non-hydrostatic modeling is favorable for the preservation of advantages of hydrostatic AREM. The non-hydrostatic and hydrostatic frames agree well with each other in terms of governing equations and modeling results when the non-hydrostatic core is degraded to the hydrostatic one. In part Ⅱ of this paper, the non-hydrostatic AREM will be verified through idealized and real-data numerical experiments.