Anomalous　(non-Fourier)　heat　transport　is　no　longer　just　a　theoretical　issue　since　it　has　been　observed　experimentally　in　a　number　of　low-dimensional　nanomaterials,　such　as　SiGe　nanowires,　carbon　nanotubes,　and　others.　To　understand　these　anomalous　behaviors,　exploring　the　microscopic　origin　of　normal　(Fourier)　heat　transport　is　a　fascinating　theoretical　topic.　However,　this　issue　has　not　yet　been　fully　understood　even　for　one-dimensional　(1D)　model　chains,　in　spite　of　a　great　amount　of　thorough　studies　done　to　date.　From　those　studies,　it　has　been　widely　accepted　that　the　conservation　of　momentum　is　a　key　ingredient　to　induce　anomalous　heat　transport,　while　momentum-nonconserving　systems　usually　support　normal　heat　transport　where　Fourier＇s　law　is　valid.　But　if　the　nonconservation　of　momentum　is　the　reason,　what　is　the　underlying　microscopic　mechanism　for　the　observed　normal　heat　transport?　Here　we　carefully　revisit　a　typical　1D　momentum-nonconserving　φ4　model,　and　we　present　evidence　that　the　mobile　discrete　breathers,　or,　in　other　words,　the　moving　intrinsic　localized　modes　with　frequency　components　above　the　linear　phonon　band,　can　be　responsible　for　that.　©　2017　American　Physical　Society.