The present disclosure relates to a process by which an optical fiber array or a single optical fiber is cleaved with a laser-cleaving apparatus. The coating material is stripped or removed from a section of an optical fiber array or single optical fiber; a coated or ribbonized section of the optical fiber array or the single optical fiber is secured in a holder; the holder is aligned inside the laser-cleaving apparatus; the laser cleaves the stripped ends of the fibers in the optical fiber array or the single optical fiber; the laser-cleaved ends of the optical fiber(s) are then mechanically separated to remove the free ends from the optical fibers in the optical fiber array or the single optical fiber, leaving a cleaved array of optical fibers or a single cleaved optical fiber. The cleaving process enables the optical fiber array or single optical fiber to be cleaved at flexible locations along an optical fiber ribbon, optical fiber, or optical fiber apparatus (e.g., cleaving can be performed close to a ferrule end face) with no swelling, minimal cleave angle variation across the cores of the optical fibers, a controlled surface roughness of the optical fiber end faces, and high process yield.