The mammalian cytoskeleton forms a mechanical continuum that spans across the cell, connecting the cell surface to the nucleus via transmembrane protein complexes in the plasma and nuclear membranes. It transmits extracellular forces to the cell interior, providing mechanical cues that influence cellular decisions, but also actively generates intracellular forces, enabling the cell to probe and remodel its tissue microenvironment. Cells adapt their gene expression profile and morphology to external cues provided by the matrix and adjacent cells as well as to cell-intrinsic changes in cytoplasmic and nuclear volume. The cytoskeleton is a complex filamentous network of three interpenetrating structural proteins: actin, microtubules, and intermediate filaments. Traditionally the actin cytoskeleton is considered the main contributor to mechanosensitivity. This view is now shifting owing to the mounting evidence that the three cytoskeletal filaments have interdependent functions due to cytoskeletal crosstalk, with intermediate filaments taking a central role. In this Mini Review we discuss how cytoskeletal crosstalk confers mechanosensitivity to cells and tissues, with a particular focus on the role of intermediate filaments. We propose a view of the cytoskeleton as a composite structure, in which cytoskeletal crosstalk regulates the local stability and organization of all three filament families at the sub-cellular scale, cytoskeletal mechanics at the cellular scale, and cell adaptation to external cues at the tissue scale.