A vertically Lagrangian finite-volume dynamical core for global models is described, featuring a terrain-following Lagrangian control-volume discretization. This approach reduces the physical problem's dimensionality from three to two, closely resembling the shallow water system. The algorithm uses a genuinely conservative flux-form semi-Lagrangian method for horizontal-to-Lagrangian-surface transport and dynamics. Time marching is split-explicit, with large time steps for scalar transport and small steps for Lagrangian dynamics, enabling accurate wave propagation. A mass, momentum, and total energy conserving algorithm is developed for remapping state variables between floating Lagrangian control-volumes and Eulerian terrain-following coordinates. Deterministic baroclinic wave growth tests and long-term integrations using Held–Suarez forcing are presented. The monotonicity constraint's impact is discussed. The core was initially developed at NASA Goddard Space Flight Center, with applications in atmospheric modeling. It builds on 1D finite-volume algorithms and extends them to multidimensional, eliminating directional splitting. The FFSL algorithm, developed in 1994, is a milestone, solving the Pole-Courant number problem and providing a stable, oscillation-free scheme. It is used in atmospheric chemistry transport models. The core was further developed by adapting FFSL to the shallow water framework, using a two-grid two-step "reversed engineering approach" to achieve consistent transport of mass, vorticity, and potential vorticity. The algorithm uses a sigma vertical coordinate, requiring a 3D transport algorithm. A simplification reduced computational cost, but the Lagrangian control-volume vertical discretization reduced dimensionality to two, making the scheme computationally efficient. The governing equations for the hydrostatic atmosphere are presented in appendix A, using a general vertical coordinate. The Lagrangian control-volume vertical discretization reduces all prognostic equations to 2D, vertically decoupled. The horizontal transport process is discretized using a 2D flux-form semi-Lagrangian algorithm, with a mass, momentum, and total energy conserving remapping algorithm for state variable remapping. Idealized tests, including baroclinic instability growth and Held–Suarez forcing simulations, demonstrate the core's accuracy and efficiency. The core is implemented in two general circulation models and is being integrated into the Geophysical Fluid Dynamics Laboratory's Flexible Modeling System. The core's performance is validated through numerical weather prediction experiments, showing significant improvements over previous systems. The core's numerical formulation can be further improved, particularly in horizontal grid choice, computational efficiency, and monotonicity constraint application. The total energy is conserved exactly through the vertical Lagrangian discretization, with the horizontal discretization using a monotonicity-preserving transport scheme. The core's performance is validated through various tests, showing its effectiveness in global modeling.A vertically Lagrangian finite-volume dynamical core for global models is described, featuring a terrain-following Lagrangian control-volume discretization. This approach reduces the physical problem's dimensionality from three to two, closely resembling the shallow water system. The algorithm uses a genuinely conservative flux-form semi-Lagrangian method for horizontal-to-Lagrangian-surface transport and dynamics. Time marching is split-explicit, with large time steps for scalar transport and small steps for Lagrangian dynamics, enabling accurate wave propagation. A mass, momentum, and total energy conserving algorithm is developed for remapping state variables between floating Lagrangian control-volumes and Eulerian terrain-following coordinates. Deterministic baroclinic wave growth tests and long-term integrations using Held–Suarez forcing are presented. The monotonicity constraint's impact is discussed. The core was initially developed at NASA Goddard Space Flight Center, with applications in atmospheric modeling. It builds on 1D finite-volume algorithms and extends them to multidimensional, eliminating directional splitting. The FFSL algorithm, developed in 1994, is a milestone, solving the Pole-Courant number problem and providing a stable, oscillation-free scheme. It is used in atmospheric chemistry transport models. The core was further developed by adapting FFSL to the shallow water framework, using a two-grid two-step "reversed engineering approach" to achieve consistent transport of mass, vorticity, and potential vorticity. The algorithm uses a sigma vertical coordinate, requiring a 3D transport algorithm. A simplification reduced computational cost, but the Lagrangian control-volume vertical discretization reduced dimensionality to two, making the scheme computationally efficient. The governing equations for the hydrostatic atmosphere are presented in appendix A, using a general vertical coordinate. The Lagrangian control-volume vertical discretization reduces all prognostic equations to 2D, vertically decoupled. The horizontal transport process is discretized using a 2D flux-form semi-Lagrangian algorithm, with a mass, momentum, and total energy conserving remapping algorithm for state variable remapping. Idealized tests, including baroclinic instability growth and Held–Suarez forcing simulations, demonstrate the core's accuracy and efficiency. The core is implemented in two general circulation models and is being integrated into the Geophysical Fluid Dynamics Laboratory's Flexible Modeling System. The core's performance is validated through numerical weather prediction experiments, showing significant improvements over previous systems. The core's numerical formulation can be further improved, particularly in horizontal grid choice, computational efficiency, and monotonicity constraint application. The total energy is conserved exactly through the vertical Lagrangian discretization, with the horizontal discretization using a monotonicity-preserving transport scheme. The core's performance is validated through various tests, showing its effectiveness in global modeling.