The cytoplasm of vertebrate cells contains three distinct filamentous biopolymers, the microtubules, microfilaments, and intermediate filaments. The basic structural elements of these three filaments are linear polymers of the proteins tubulin, actin, and vimentin or another related intermediate filament protein, respectively. The viscoelastic properties of cytoplasmic filaments are likely to be relevant to their biologic function, because their extreme length and rodlike structure dominate the rheologic behavior of cytoplasm, and changes in their structure may cause gel-sol transitions observed when cells are activated or begin to move. This paper describes parallel measurements of the viscoelasticity of tubulin, actin, and vimentin polymers. The rheologic differences among the three types of cytoplasmic polymers suggest possible specialized roles for the different classes of filaments in vivo. Actin forms networks of highest rigidity that fluidize at high strains, consistent with a role in cell motility in which stable protrusions can deform rapidly in response to controlled filament rupture. Vimentin networks, which have not previously been studied by rheologic methods, exhibit some unusual viscoelastic properties not shared by actin or tubulin. They are less rigid (have lower shear moduli) at low strain but harden at high strains and resist breakage, suggesting they maintain cell integrity. The differences between F-actin and vimentin are optimal for the formation of a composite material with a range of properties that cannot be achieved by either polymer alone. Microtubules are unlikely to contribute significantly to interphase cell rheology alone, but may help stabilize the other networks.