The cytoskeleton comprises networks of different biopolymers, which serve various cellular functions. To accomplish these tasks, their mechanical properties are of particular importance. Understanding them requires detailed knowledge of the mechanical properties of the individual filaments that make up these networks, in particular, microtubules, actin filaments, and intermediate filaments. Far from being homogeneous beams, cytoskeletal filaments have complex mechanical properties, which are directly related to the specific structural arrangement of their subunits. They are also versatile, as the filaments' mechanics and biochemistry are tightly coupled, and their properties can vary with the cellular context. In this review, we summarize decades of research on cytoskeletal filament mechanics, highlighting their most salient features and discussing recent insights from this active field of research.