This paper describes a finite-volume dynamical core for global models that uses a terrain-following Lagrangian control-volume discretization, reducing the dimensionality of the physical problem from three to two. The 2D horizontal transport and dynamical processes are discretized using a conservative flux-form semi-Lagrangian algorithm. Time marching is split-explicit, with large time steps for scalar transport and small fractional steps for Lagrangian dynamics, allowing accurate propagation of fast waves. A mass, momentum, and total energy conserving remapping algorithm is developed to periodically remap state variables from the floating Lagrangian control-volume to an Eulerian terrain-following coordinate. The paper presents deterministic baroclinic wave-growth tests and long-term integrations using the Held–Suarez forcing, discussing the impact of the monotonicity constraint. The finite-volume dynamical core has been successfully implemented into two general circulation modeling systems and is being developed for unstructured grids.This paper describes a finite-volume dynamical core for global models that uses a terrain-following Lagrangian control-volume discretization, reducing the dimensionality of the physical problem from three to two. The 2D horizontal transport and dynamical processes are discretized using a conservative flux-form semi-Lagrangian algorithm. Time marching is split-explicit, with large time steps for scalar transport and small fractional steps for Lagrangian dynamics, allowing accurate propagation of fast waves. A mass, momentum, and total energy conserving remapping algorithm is developed to periodically remap state variables from the floating Lagrangian control-volume to an Eulerian terrain-following coordinate. The paper presents deterministic baroclinic wave-growth tests and long-term integrations using the Held–Suarez forcing, discussing the impact of the monotonicity constraint. The finite-volume dynamical core has been successfully implemented into two general circulation modeling systems and is being developed for unstructured grids.