Function-on-function　linear　regression　is　an　essential　tool　in　characterizing　the　linear　relationship　between　a　functional　response　and　a　functional　predictor.　However,　most　of　the　estimation　methods　for　this　model　are　based　on　the　least-squares　procedure,　which　is　sensitive　to　atypical　observations.　In　this　paper,　we　present　a　robust　method　for　the　function-on-function　linear　model　using　M-estimation　and　penalized　spline　regression.　A　fast　iterative　algorithm　is　provided　to　compute　the　estimates.　The　efficiency　of　the　proposed　robust　penalized　M-estimator　is　investigated　with　several　simulation　studies　in　comparison　with　the　conventional　method.　We　demonstrate　the　performance　of　the　proposed　robust　method　with　two　real　data　examples　in　a　capital　bike-sharing　study　and　a　Hawaii　ocean　time-series　program.