Heavy rain will bring the risk of localised flooding in places.
Travellers have been warned to brace themselves for disruption, particularly on Tuesday.
After a chilly but sunny start for most, much of the UK will stay dry on Sunday with bright or sunny spells continuing throughout.
It is the fifth year that the Met Office and Met Eireann have run the Name our Storms campaign.
The showers will be most frequent in the north and west of the country.
The Met Office has already received hundreds of suggestions of what to call storms for the upcoming season.
Britain’s highest ever temperature record could fall on Thursday.
The dry weather precedes a four-day heatwave next week when temperatures could peak at 34C.
It's a fresher day but we could still see a high of 25 degrees.
It will be a warm day for all, with top temperatures of 26 Celsius.