Eukaryotic chemotaxis involves distinct cell shapes, with movement in shallow gradients dominated by split pseudopods and a single, broad leading edge in steep gradients, but little is known about the significance of these modes. In this thesis, I demonstrate that the shape of aggregating Dictyostelium discoideum cells is important for chemotaxis at the fundamental limit of gradient sensing. Using Fourier shape descriptors, I show that Dictyostelium cells occupy a naturally low-dimensional space of shapes, and that these cell shapes depend on the external environment. I present evidence that this space is restricted by treatments with a phospholipase A2 inhibitor, which is known to inhibit chemotaxis. I show that biophysical simulations can recreate wild-type chemotaxis and shape behaviour, and that restrictions to the shape of these simulations alone, with no change made to their biochemistry, are sufficient to recreate the drop in chemotactic accuracy seen in drug-treated live cells. I then discuss further applications of physical principles to understanding cell shape and chemotaxis, and the application of shape analysis to other areas of cell biology, specifically to the formation of immune synapses in T cells.