The cytoskeleton is generally visualized by light or electron microscopy as a meshwork of protein filaments that spans the space between the nuclear envelope and the plasma membrane. In most cell types, this meshwork is formed by a three dimensional composite network of actin filaments, microtubules (MT), and intermediate filaments (IF) together with the host of proteins that bind to the sides or ends of these linear polymers. Cytoskeletal binding proteins regulate filament length, crosslink filaments to each other, and apply forces to the filaments. One approach to modeling the mechanical properties of the cytoskeleton and of cell in general is to consider the elements of the cytoskeleton as polymers, using experimental methods and theoretical models developed for traditional polymers but modified for the much larger, stiffer, and fragile biopolymers comprising the cytoskeleton. The presence of motor proteins that move actin filaments and microtubules also creates a new class of active materials that are out of thermodynamic equilibrium, and unconstrained by limitations of the fluctuation-dissipation theorem. These active materials create rich opportunities for experimental design and theoretical developments. The degree to which the mechanics of live cells can usefully be modeled as highly complex polymer networks is by no means certain, and this article will discuss recent progress in quantitatively measuring cytoskeletal polymer systems and relating them to the properties of the cell.