We provide a review of recent advancements in non-local continuous models for migration, mainly from the perspective of its involvement in embryonal development and cancer invasion. Particular emphasis is placed on spatial non-locality occurring in advection terms, used to characterize a cell’s motility bias according to its interactions with other cellular and acellular components in its vicinity (e.g. cell–cell and cell–tissue adhesions, non-local chemotaxis), but we also briefly address spatially non-local source terms. Following a short introduction and description of applications, we give a systematic classification of available PDE models with respect to the type of featured non-localities and review some of the mathematical challenges arising from such models, with a focus on analytical aspects.

This article is part of the theme issue ‘Multi-scale analysis and modelling of collective migration in biological systems’.

我们回顾了非局部连续迁移模型的最新进展，主要是从其参与胚胎发育和癌症侵袭的角度来看。特别强调平流术语中发生的空间非定域性，用于根据细胞与其附近其他细胞和非细胞成分的相互作用（例如细胞-细胞和细胞-组织粘附、非局部趋化性）来表征细胞的运动偏向性，但我们也简要介绍了空间上的非本地源术语。在对应用程序进行简短介绍和描述之后，我们根据特征非定域的类型对可用的 PDE 模型进行了系统分类，并回顾了这些模型带来的一些数学挑战，重点是分析方面。

本文是主题问题“生物系统中集体迁移的多尺度分析和建模”的一部分。