The cancelled services will resume from approximately 10am this morning.
However, some routes are currently running, albeit with minor disruption. These include services between Edinburgh and Dunbar, Edinburgh and North Berwick, Edinburgh and Glasgow Queen Street via Falkirk High, and Edinburgh and Glasgow Queen Street via Cumbernauld.
Trains between Edinburgh and Dunblane are now returning to run as normal.
Scotrail engineers are currently checking routes and assessing the damage caused by the storm.
These cancellations come after Storm Dudley struck the Central Belt yesterday evening.
The Met Office had issued an amber warning for strong winds covering central Scotland, and a yellow wind warnings for the rest of Scotland – excluding Shetland.
Caledonian MacBrayne also announced that several ferry services were liable to disruption or cancellation at short notice.
Historic Scotland announced the early closure of Edinburgh Castle on Wednesday afternoon, and have delayed opening the castle until 10.30am this morning.
Yesterday’s weather will be followed on Friday by Storm Eunice which could be even more damaging, with stronger winds than Dudley, heavy snow and possible blizzard conditions.
The Met Office have issued a yellow warning for wind and snow, which covers Edinburgh and the Lothians. This alert will come into effect at 3am on Friday and last until 6pm.