Movement of individual cells and of cellular cohorts, chains or sheets requires physical forces that are established through interactions of cells with their environment. In vivo, migration occurs extensively during embryonic development and in adults during wound healing and tumorigenesis. In order to identify the molecular events involved in cell movement, in vitro systems have been developed. These have contributed to the definition of a number of molecular pathways put into play in the course of migratory behaviours, such as mesenchymal and amoeboid movement. More recently, our knowledge of migratory modes has been enriched by analyses of cells exploring and moving through three-dimensional (3D) matrices. While the cells’ morphologies differ in 2D and 3D environments, the basic mechanisms that put a cellular body into motion are remarkably similar. Thus, in both 2D and 3D, the polarity of the migrating cell is initially defined by a specific subcellular localization of signalling molecules and components of molecular machines required for motion. While the polarization can be initiated either in response to extracellular signalling or be a chance occurrence, it is reinforced and sustained by positive feedback loops of signalling molecules. Second, adhesion to a substratum is necessary to generate forces that will propel the cell engaged in either mesenchymal or ameboid migration. For collective cell movement, intercellular coordination constitutes an additional requirement: a cell cohort remains stationary if individual cells pull in opposite directions. Finally, the availability of space to move into is a general requirement to set cells into motion. Lack of free space is probably the main obstacle for migration of most healthy cells in an adult multicellular organism. Thus, the requirements for cell movement are both intrinsic to the cell, involving coordinated signalling and interactions with molecular machines, and extrinsic, imposed by the physicochemical nature of the environment. In particular, the geometry and stiffness of the support act on a range of signalling pathways that induce specific cell migratory responses. These issues are discussed in the present review in the context of published work and our own data on collective migration of hepatocyte cohorts.