Additive Manufacturing (AM) via Laser Powder Bed Fusion (LPBF) is often described as a rapid solidification process due to its high cooling rates. However, clear evidence of far-from-equilibrium solidification phenomena – such as solute trapping, kinetic undercooling, or morphological transitions – remains limited in LPBF-processed alloys. In this talk, we revisit whether fusion-based AM and LPBF truly qualify as rapid solidification processes and illustrate how phase-field (PF) models, which capture deviations from thermodynamic equilibrium at the solid-liquid interface, can shed light on microstructure selection mechanisms in metallic AM. We first examine LPBF-processed WE43 biomedical Mg alloy, displaying "banded" microstructures characteristic of rapid solidification. Coupling melt pool-scale thermal modeling with PF simulations confirms that thermal conditions in LPBF promote a banding instability, driven by strong kinetics-induced deviations from equilibrium at the solid-liquid interface. [Modell. Sim. Mater. Sci. Eng. 32 (2024) 055012] Next, we investigate the broader occurrence of this banding instability, which has been observed in a range of Al alloys but is notably absent in well-studied systems like Ni- and Fe-based alloys. A parametric analysis highlights the influence of material properties – for instance the solid-liquid interface kinetic coefficient – on the emergence or suppression of this instability, which lies between the cellular/dendritic regime and the high-velocity absolute planar stability regime. The greater fundamental understanding and predictive capabilities provided by quantitative PF models of rapid solidification open exciting avenues for the design of alloys, the optimization of processes, and the manufacturing of unique outstanding parts and components, e.g. functionally graded materials.