Cytosolic fluid dynamics have been implicated in cell motility because of the hydrodynamic forces they induce and because of their influence on transport of components of the actin machinery to the leading edge. To investigate the existence and the direction of fluid flow in rapidly moving cells, we introduced inert quantum dots into the lamellipodia of fish epithelial keratocytes and analysed their distribution and motion. Our results indicate that fluid flow is directed from the cell body towards the leading edge in the cell frame of reference, at about 40% of cell speed. We propose that this forward-directed flow is driven by increased hydrostatic pressure generated at the rear of the cell by myosin contraction, and show that inhibition of myosin II activity by blebbistatin reverses the direction of fluid flow and leads to a decrease in keratocyte speed. We present a physical model for fluid pressure and flow in moving cells that quantitatively accounts for our experimental data.