Here we review studies of the physical, material properties of animal cells and their cytoskeleton, such as elastic stiffness and fluid viscosity, that determine how they respond to, and are shaped by, forces inside and out. Currently and historically, most such studies have reported a single value for a cell property and/or propose a single broad structural model based on nonliving materials. We believe that such physical studies would be of more interest to most cell biologists if greater emphasis were placed on the well-established regional differences within a cell and the ability of the cell to quickly change its mechanical behaviors.