This review article summarizes the main outcomes following from recently developed theories of stable dendritic growth in undercooled one-component and binary melts. The nonlinear heat and mass transfer mechanisms that control the crystal growth process are connected with hydrodynamic flows (forced and natural convection), as well as with the non-local diffusion transport of dissolved impurities in the undercooled liquid phase. The main conclusions following from stability analysis, solvability and selection theories are presented. The sharp interface model and stability criteria for various crystallization conditions and crystalline symmetries met in actual practice are formulated and discussed. The review is also focused on the determination of the main process parameters—the tip velocity and diameter of dendritic crystals as functions of the melt undercooling, which define the structural states and transitions in materials science (e.g. monocrystalline-polycrystalline structures). Selection criteria of stable dendritic growth mode for conductive and convective heat and mass fluxes at the crystal surface are stitched together into a single criterion valid for an arbitrary undercooling.

This article is part of the theme issue ‘Transport phenomena in complex systems (part 1)’.

这篇评论文章总结了最近发展的过冷单组分和二元熔体中稳定枝晶生长理论的主要结果。控制晶体生长过程的非线性传热和传质机制与流体动力学流动（强制对流和自然对流）以及过冷液相中溶解杂质的非局部扩散传输有关。提出了从稳定性分析、可解性和选择理论得出的主要结论。制定并讨论了在实际实践中满足的各种结晶条件和晶体对称性的清晰界面模型和稳定性标准。该评论还侧重于确定主要工艺参数 - 作为熔体过冷度的函数的树枝状晶体的尖端速度和直径，它们定义了材料科学中的结构状态和转变（例如单晶-多晶结构）。晶体表面传导和对流热和质量通量的稳定枝晶生长模式的选择标准被拼接成一个对任意过冷有效的单一标准。

本文是主题问题“复杂系统中的传输现象（第 1 部分）”的一部分。