Storm Atiyah: South West battered by 73mph winds
People were warned to stay indoors and avoid the roads unless necessary after gusts of more than 70mph (112km/h) battered parts of the UK overnight.
Cornwall Fire and Rescue Service issued the advice as it dealt with a number of storm-related incidents. A similar warning was issued on Jersey.
Hundreds of homes in Devon and Cornwall were without power.
In Bude, a road was closed after part of a Sainsbury's supermarket roof came off on Sunday night.
The supermarket said the store remained closed on Monday morning but staff were working hard to reopen. The spokesman said the popular Bude supermarket tunnel, which topped TripAdvisor ratings last year, was safe.
There were reports of fallen trees blocking a number of roads in Cornwall, where winds reached speeds of 73 mph (117 kmph), while Highways England said 29 trees came down in Devon overnight.
The A377 at Umberleigh, Devon, was 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 were blocking lines between Penzance and Plymouth.
Ferry services including the Wightlink, Brittany Ferries and and Hovertravel were also disrupted, with some services suspended or cancelled.
Wales suffered similar problems as a result of the storm, with 1,300 homes left without power.
Gusts of 77mph (120km/h) hit Aberdaron on the Llŷn Peninsula in Gwynedd and 74mph (119km/h) in Aberporth, Ceredigion.
Police warned motorists "multiple" trees had blocked roads in mid and west Wales and the weather meant restrictions had to be put in place on several bridges.
Ferry services between Dublin and Holyhead were also disrupted.
People in the Channel Islands were being urged to take extra care and "avoid outside activity" as Jersey Met issued a red storm warning.
The Met Office's weather warning for Storm Atiyah was in place until 09:00 GMT. Winds were expected to become much lighter by the end of the afternoon.