A wire travelling device including a wire-feed-roller circumferential-speed calculation unit and a wire-collection roller circumferential-speed calculation unit that calculate each circumferential speed of the wire feed roller and the wire collection roller, a circumferential-speed-difference holding unit that calculates a difference between the circumferential speed of the wire feed roller and the circumferential speed of the wire collection roller as a circumferential speed difference and holds therein the calculated circumferential speed difference, and a wire-feed-roller speed control unit and a wire-collection roller speed control unit that control the wire feed roller and the wire collection roller, in changing a traveling speed of the wire, by a wire-feed-roller speed instruction value and a wire-collection roller speed instruction value generated so that the circumferential speed difference before changing the traveling speed of the wire is maintained after the changing.