People are being warned to stay indoors and not take to the roads unless necessary after gusts of up to 73mph (117km/h) were recorded in Cornwall overnight.
Cornwall Fire and Rescue Service issued the advice and said it was also dealing with a number of incidents.
In Bude, Crooklets Road is closed after part of a Sainsbury’s supermarket roof came off on Sunday night.
Hundreds of homes in Devon and Cornwall are without power.
There are reports of fallen trees blocking a number of roads in Cornwall.
The A377 at Umberleigh, Devon, has been closed by a landslip and several large fallen trees.
In another incident at Topsham, Devon, a police officer tweeted photos of a colleague lifting a tree out of the way of a dark road, writing: “Topsham you are most welcome.”
The high winds also caused disruption to trains and ferries, with National Rail warning several trees are blocking lines between Penzance and Plymouth.
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Ferry services including the Wightlink, Brittany Ferries and and Hovertravel are also disrupted, with some services suspended or cancelled.
The Met Office’s weather warning for Storm Atiyah was due to remain in place until 09:00 GMT.