Cells and tissues are mechanical as well as biochemical machines, and cellular response to mechanical cues can have as large an influence on structure and function as chemical signals. The mechanical properties of cells are largely determined by networks of semiflexible polymers forming the cytoskeleton, which has viscoelastic properties that differ in important ways from the viscoelasticity of common synthetic materials. Two such features are the high resistance to deformation achieved by a remarkably low volume fraction of protein, and the increase in stiffness that occurs when the cytoskeletal network is deformed. The actin filaments, microtubules and intermediate filaments that comprise the cytoskeleton of most cell types are linear polymers with some important similarities but also some fundamental differences. The stiffness of the individual polymer types is vastly different, with persistence lengths ranging from 1 mm for the 24 nm diameter microtubules to a few 100 nm for the 10-14 nm diameter intermediate filaments. The material properties of these biopolymer networks are proposed to function as part of the mechanosensing mechanism in cells, and the stiffness of cytoskeletal networks is similar to that of common extracellular protein networks such as those formed by collagen and fibrin in which many cell types function. Examples of the morphologic differences in fibroblasts and astrocytes grown on chemically identical surfaces overlying gels with elastic moduli spanning the range from 50 to 12,000 Pa illustrate the large effect of stiffness differences on cell structure and function.