Bubble domain (BD) motion is expected to be useful for domain motion magnetic memory. We investigate the current-induced motion of BDs produced in a perpendicularly magnetised thin wire with edge defects by using micro-magnetic simulation. We produce a stable BD in the wire and find that its size increases with the wire width. Upon application of a spin-polarised current, the BD both with and without Bloch points moves with a velocity similar to the theoretical velocity of domain wall motion while maintaining its structure. In the presence of edge defects in the wire, the motion of the BD is slightly affected by the magnetisation of the uneven edge structure, whereas a straight domain wall is pinned strongly by the edge defects. The effect of edge defects on BD motion is much smaller than that in straight domain wall motion.