Some train services and ferry sailings have also been affected by the storm, with Holyhead Port currently closed and the scheduled sailings to Dublin cancelled “due to infrastructure weather damage”.
Transport for Wales (TfW) said all railway lines are blocked on 11 routes, including those between Swansea and Milford Haven, Swansea and Shrewsbury and Chester and Holyhead.
TfW has also warned people to check before they travel on a number of routes.
National Rail Enquiries has warned that “services may be busier than normal today and experience severe overcrowding”.
Great Western Railway said passengers should “not attempt to travel” between Swansea and Carmarthen until at least 12:00 GMT.
A number of schools across Wales are closed or partially closed, including three in Bridgend, external, five in Carmarthenshire, external, four in Powys, external, two in Gwynedd, external and one in Flintshire, external.