Abstract:  In order to fight infection or heal injury, living cells have to be able to move in response to chemical cues around them. As a first step, an internal chemical "map" is rapidly induced in the cell, leading to its polarization, reorganization of structural proteins (cytoskeleton), shape change, and crawling motility. In my talk, I will summarize some of the work done in my group addressing these processes. I will explain the basic properties of the signalling proteins (small GTPases) and their role in cell polarization. I will motivate a sequence of mathematical models that we have studied to understand the underlying mechanisms. One theme in my talk will be the cross-fertilization of the complex biological problem and its simplified mathematical caricatures. Another theme is the effect of cell shape on internal chemical dynamics. I will conclude with recent developments on computations of cell shape dynamics in 2D simulations.