The cytoskeleton of eukaryotic cells is primarily composed of networks of filamentous proteins, F-actin, microtubules, and intermediate filaments. Interactions among the cytoskeletal components are important in determining cell structure and in regulating cell functions. For example, F-actin and microtubules work together to control cell shape and polarity, while the subcellular organization and transport of vimentin intermediate filament (VIF) networks depend on their interactions with microtubules. However, it is generally thought that F-actin and VIF5 form two coexisting but separate networks that are independent due to observed differences in their spatial distribution and functions. In this paper, we present a closer investigation of both the structural and functional interplay between the F-actin and VIF cytoskeletal networks. We characterize the structure of VIF5 and F-actin networks within the cell cortex using structured illumination microscopy and cryoelectron tomography. We find that VIF5 and F-actin form an interpenetrating network (IPN) with interactions at multiple length scales, and VIF5 are integral components of F-actin stress fibers. From measurements of recovery of cell contractility after transient stretching, we find that the IPN structure results in enhanced contractile forces and contributes to cell resilience. Studies of reconstituted networks and dynamic measurements in cells suggest direct and specific associations between VIF5 and F-actin. From these results, we conclude that VIF5 and F-actin work synergistically, both in their structure and in their function. These results profoundly alter our understanding of the contributions of the components of the cytoskeleton, particularly the interactions between intermediate filaments and F-actin.