No trains are running between London Paddington and Bristol or South Wales due to flooding on the Western line.
Network Rail said "severe disruption" was expected to last for the rest of today and passengers should make "alternative arrangements".
Roads are also closed across the west country as Storm Claudia brings amber weather warnings and heavy downpours.
Traffic monitoring site INRIX is reporting queueing traffic and one lane closed due to flooding on the M5 southbound after J19 A369 near Portishead, Somerset.
A large Met Office yellow severe weather warning is in force across most of England and Wales and will be valid until Saturday at 06:00 GMT.
"Routes between London Paddington and Swansea via Bristol Parkway and London Paddington to Bristol Temple Meads via Swindon are currently closed owing to flooding," said Network Rail.
"Passengers are advised not to attempt to travel on these routes."
It said tickets dated for Friday would be accepted on services on Saturday and Sunday
Meanwhile, drivers on the M5 northbound have also faced delays of more than two hours as persistent rain, a road defect and a car crash have caused standstill traffic between J20 and J19, in North Somerset.
National Highways is warning of "severe delays" of 140 minutes.
In Banwell, North Somerset, congestion is building on the A371 West Street in both directions, from Knightcott Gardens to Westfield Crescent, as flooding has rendered the road "impassable", according to INRIX.
Great Western Railway (GWR) has closed the line between Chippenham and Bath Spa, affecting services between Swindon and Westbury via Melksham.
Network Rail is monitoring the water levels, but until the water recedes and the line is inspected, trains will not be running in either direction, a GWR spokesperson said.
The railway line from Gloucester to Bristol Parkway is also closed.
This will affect services in the other direction too, as trains that would form services from Bristol Temple Meads will not have been able to run down from Gloucester.
Routes from London Paddington to Bristol Temple Meads via Swindon are currently closed.
And in Swindon, Wiltshire, Piper's Way has been closed due to heavy flooding.
In a statement, Swindon Borough Council said: "While our team have cleared more than 13,000 gullies across the borough this year, Swindon's culverts and drains can still struggle with such a high volume of water in a short period of time.
"The road will be reopened once it is safe to do so and the flooding has subsided."
A government warning for "imminent flooding" has been issued for homes in South Marston Brook in Swindon.
Properties closest to the river, including Quarrybrook Close, Byron Court, Greenfields and Chapel Lane, are most at risk.
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