Sensing of damage, deformation, and mechanical forces is of vital importance in many applications of fiber-reinforced polymer composites, as it allows the structural health and integrity of composite components to be monitored and microdamage to be detected before it leads to catastrophic material failure. Bioinspired and biomimetic approaches to self-sensing and self-reporting materials are reviewed. Examples include bruising coatings and bleeding composites based on dye-filled microcapsules, hollow fibers, and vascular networks. Force-induced changes in color, fluorescence, or luminescence are achieved by mechanochromic epoxy resins, or by mechanophores and force-responsive proteins located at the interface of glass/carbon fibers and polymers. Composites can also feel strain, stress, and damage through embedded optical and electrical sensors, such as fiber Bragg grating sensors, or by resistance measurements of dispersed carbon fibers and carbon nanotubes. Bioinspired composites with the ability to show autonomously if and where they have been damaged lead to a multitude of opportunities for aerospace, automotive, civil engineering, and wind-turbine applications. They range from safety features for the detection of barely visible impact damage, to the real-time monitoring of deformation of load-bearing components.