Amber and yellow warnings are in place for northern England, Scotland and Northern Ireland, where winds up to 90mph are expected to hit on Wednesday and Thursday.
A yellow wind warning has also been issued for the lower part of Scotland and the rest of the UK for Friday, when gusts could get up to 80mph.
The Storm Dudley yellow alert comes into force for the north of the UK at 3pm on Wednesday and will last until 6pm on Thursday.
An amber alert will also be in force between 6pm on Wednesday and 9am on Thursday for the southern part of Scotland, as well as northern England and Northern Ireland.
Storm Eunice is likely to hit on Friday, when a yellow alert will be in force until 9pm.
The Met Office has warned flying debris could cause injuries and danger to life, while large waves also pose a risk and beach material could be lifted up and thrown onto seafronts, coastal roads and properties.
Severe travel disruption is also likely, with road, rail, air and ferry services potentially affected, and some parts could be affected by power cuts.