This critical review provides the first coherent elucidation of the science behind electrical-resistance-based strain/damage/temperature self-sensing in continuous carbon fiber polymer-matrix composites, which are important for lightweight structures (e.g., aircraft). Self-sensing pertains to smart structures. It is based on structure–property relationships. There is no device incorporation. The type of loading and the type of resistance measurement affect the resistance change. The resistance changes are consistently elucidated in terms of the structure–property relationships. Upon elastic flexure, the tension-surface resistance increases, due to the fiber waviness decrease and consequent current penetration decrease, while the compression-surface resistance decreases. Upon elastic through-thickness compression, the through-thickness resistivity decreases, due to the enhanced through-thickness fiber proximity, thus promoting the longitudinal current path detour necessitated by fiber imperfection and decreasing the longitudinal resistivity. For a single fiber in epoxy, the resistivity decreases upon tension, due to the decrease of the composite-fabrication-induced residual compressive stress in the fiber. For multi-lamina laminates, the residual compressive stress decrease upon elastic longitudinal tension causes the longitudinal resistivity to decrease, while the through-thickness resistivity increases due to the fiber waviness decrease. For 1-lamina laminates, elastic longitudinal tension decreases the relatively limited fiber waviness, thereby increasing the through-thickness/transverse resistivity and consequently hindering the longitudinal current path detour and increasing the longitudinal resistivity. Delamination decreases the interlaminar fiber touching, thereby increasing the through-thickness/interlaminar resistivity. Heating decreases the interlaminar resistivity, due to interlaminar electron jumping, thereby enabling temperature sensing. Fiber fracture and interlaminar degradation increase the resistance irreversibly. For sensing matrix cracking, conductive filler incorporation helps.