A cleaving apparatus holds at least one surface (effective surface) of a band-like glass film in a non-contact state to suppress a situation that wavy portions reach a region in which the band-like glass film is to be cleaved. The cleaving apparatus cleaves the band-like glass film, which is being conveyed in a longitudinal direction thereof, along a conveyance direction thereof using a thermal stress generated through localized heating and cooling of a heated region. The localized heating and the cooling is performed on a preset cleaving line extending along the conveyance direction of the band-like glass film. The cleaving apparatus includes an air knife for supplying air to a front surface of the band-like glass film to retain the film on a conveyor at a position on an upstream side of a cleaving region in the conveyance direction.