According to one embodiment, liquid crystal optical element includes: first board; second board; liquid crystal layer; first electrodes; second electrodes; third electrodes; and fourth electrode. The liquid crystal optical element generates refractive index distribution functioning as a lens. The first electrodes are at positions corresponding to end portions of the lens and on the liquid crystal layer side of the first board. The second electrodes are at positions corresponding to discontinuity points of the lens and on the liquid crystal layer side of the first board. The third electrodes are at positions corresponding to lens surfaces of the lens and on the liquid crystal layer side of the first board. The fourth electrode is provided to the whole surface on the liquid crystal layer side of the second board and includes first cutout portions formed at parts opposing the third electrodes.