In the year to June 2017, there were 505 complaints, compared with 382 in the previous year.
The figures were compiled by the UK European Consumer Centre (ECC), which helps consumers who have issues with firms based in the European Union.
The most frequent complaint was from consumers who said they had been charged extra after returning the car.
The countries which caused UK consumers the most problems were Spain, Italy, Iceland and Malta.
Many consumers find they are sent a surprise bill many weeks or months after they have returned to the UK, charging them for damage to the car.
The ECC said such bills can come as a total shock.
In some cases, the car hire firms will send photographs as proof of the damage, but these photos can be taken days after the car was returned.
Other typical complaints include:
"Our general advice to consumers hiring cars abroad is, where possible, to return the car to someone in the office, get them to check the car and sign it off as in good condition," said Andy Allen, director of the UK ECC.
"If you have to leave the car, take some photos - showing that it was returned in good condition."
The number of Britons complaining about car hire in Europe has risen by more than a third over the last 12 months.