Vapor-phase growth of snow crystals generates a striking diversity of morphologies—from simple faceted prisms to complex stellar dendrites, hollow columns, bullet rosettes, and rare trigonal forms—from a single hexagonal ice Ih lattice. This polymorphism emerges from the interplay between temperature-dependent anisotropy in facet-specific attachment kinetics, diffusion-limited vapor transport, and morphological instabilities at the ice–vapor interface. Despite extensive research, no first-principles predictive framework exists; existing models depend on empirical parameterizations of the attachment coefficient <em>α</em>(<em>T</em>) and provide limited insight into how trace atmospheric impurities alter step energetics, quasi-liquid layer (QLL) stability, and growth kinetics. Here we review two decades of advances in laboratory experiments, theory, and simulations since Libbrecht’s 2005 synthesis. We introduce a tripartite coupling framework that unifies the ice crystal, water vapor, and background atmospheric constituents (including impurities). Central to the framework is the Structure-Dependent Attachment Kinetics (SDAK) model, which accounts for habit transitions, edge-sharpening instabilities, trigonal symmetry breaking, and QLL-mediated step dynamics. We discuss extensions to climate microphysics, icephobic surface design, and planetary cryoscience, and identify key remaining challenges and future directions.