The cellular cytoskeleton is a dynamic network of filamentous proteins, consisting of filamentous actin (F-actin), microtubules, and intermediate filaments. However, these networks are not simple linear, elastic solids; they can exhibit highly nonlinear elasticity and a thermal dynamics driven by ATP-dependent processes. To build quantitative mechanical models describing complex cellular behaviors, it is necessary to understand the underlying physical principles that regulate force transmission and dynamics within these networks. In this chapter, we review our current understanding of the physics of networks of cytoskeletal proteins formed in vitro. We introduce rheology, the technique used to measure mechanical response. We discuss our current understanding of the mechanical response of F-actin networks, and how the biophysical properties of F-actin and actin cross-linking proteins can dramatically impact the network mechanical response. We discuss how incorporating dynamic and rigid microtubules into F-actin networks can affect the contours of growing microtubules and composite network rigidity. Finally, we discuss the mechanical behaviors of intermediate filaments.