The ICON (ICOsahedral Non‐hydrostatic) modelling framework of DWD and MPI‐M: Description of the non‐hydrostatic dynamical core. This article describes the non‐hydrostatic dynamical core developed for the ICOsahedral Non‐hydrostatic (ICON) modelling framework. ICON is a joint project of the German Weather Service (DWD) and the Max Planck Institute for Meteorology (MPI‐M), targeting a unified modelling system for global numerical weather prediction (NWP) and climate modelling. Compared with the existing models at both institutions, the main achievements of ICON are exact local mass conservation, mass‐consistent tracer transport, a flexible grid nesting capability and the use of non‐hydrostatic equations on global domains. The dynamical core is formulated on an icosahedral‐triangular Arakawa C grid. Achieving mass conservation is facilitated by a flux‐form continuity equation with density as the prognostic variable. Time integration is performed with a two‐time‐level predictor–corrector scheme that is fully explicit, except for the terms describing vertical sound‐wave propagation. To achieve competitive computational efficiency, time splitting is applied between the dynamical core on the one hand and tracer advection, physics parametrizations and horizontal diffusion on the other hand. A sequence of tests with varying complexity indicates that the ICON dynamical core combines high numerical stability over steep mountain slopes with good accuracy and reasonably low diffusivity. Preliminary NWP test suites initialized with interpolated analysis data reveal that the ICON modelling system already achieves better skill scores than its predecessor at DWD, the operational hydrostatic Global Model Europe (GME), and at the same time requires significantly fewer computational resources