Arriva Trains Wales says 12 trains sustained damage from hitting trees and other debris on the railway yesterday.
As a result, today, trains across the entire network could be cancelled or run with fewer carriages than usual while these damaged trains undergo essential safety checks and repairs.
The company says passengers should check services before traveling.
There is no service between Llandudno and Blaenau Ffestiniog because of a fallen tree on the line.
There is a replacement bus service running.
The same is true between Wrexham Central and Bidston, and services are also not running on the Heart of Wales line, between Llanelli and Shrewsbury.
Arriva says services are unlikely to be up and running until tomorrow.