This work presents a new shallow water moment model posed in local curvilinear coordinates intrinsically attached to a reference curve. The use of local curvilinear coordinate systems has emerged as a powerful methodology to describe flows over complex geometries, allowing geometric effects to be incorporated in a consistent manner (see, e.g. Bachini and Putti, 2020). In parallel, shallow water moment equations have been developed to enrich classical depth-averaged models by capturing the vertical structure of the flow through higher-order moments (see, e.g, Garres-Diaz et al., 2020). However, existing moment-based formulations are typically derived in Cartesian settings and therefore do not fully account for curvature-induced effects. The present work addresses this gap by combining the moment methodology with a curvilinear framework. Starting from the Navier–Stokes equations, we derive a one-dimensional shallow water system in local curvilinear coordinates and apply a moment expansion to describe the vertical velocity profile. This leads to the curvilinear shallow water moment equations (CSWME), which naturally incorporate geometric contributions while extending standard shallow water models beyond the depth-averaged level. A series of numerical experiments is used to investigate the role of curvature and friction. The results demonstrate that, for steep or strongly curved beds, curvilinear and Cartesian formulations can produce markedly different flow dynamics, with Cartesian models systematically predicting faster downslope motion. Moreover, even in frictionless configurations and starting from rest, curvature induces asymmetry in the vertical velocity profile, generating non-zero higher-order moments that are absent in Cartesian geometries. These findings highlight the importance of geometry-aware formulations and show that the CSWME provide a robust and physically consistent framework for modelling shallow flows over complex terrains with enhanced vertical resolution.