Scotrail have confirmed that services to and from the Scottish Borders will be wound down by 4pm.
The last train to leave Edinburgh for Tweedbank will leave at 3.24pm.
The last train leaving Tweedbank to Edinburgh is currently scheduled to leave at 3.29pm.
The rail company say that strength of the winds expected could damage infrastructure, blowing debris and trees onto tracks and damaging equipment such as overhead electric power lines and signals.
Network Rail will have additional engineers out across the network ready to react to problems and will check all affected lines for damage before reintroducing services as quickly as possible.
Disruption on some lines is expected to continue until mid-morning on Thursday.
Further significant weather is being forecast for most of Scotland on Friday (February 18).
A spokesperson said: “If you have to travel over the next few days, please check your journey before setting off.”
Storm Dudley, the fifth named storm of the season, commenced at 2pm, two hours earlier than previously reported.