The European Commission said the end of roaming charges was one of the greatest successes of the EU.

But a UK consumer group warned phone users could face "unexpected charges".

Until now roaming, or connection, charges have been added to the cost of calls, texts and internet browsing when consumers from one EU country travelled to another and connected to a mobile network there.

In some cases, mobile users have faced bills of hundreds of pounds if, for example, they have downloaded a film.

In a statement the European Commission said: "Each time a European citizen crossed an EU border, be it for holidays, work, studies or just for a day, they had to worry about using their mobile phones and a high phone bill from the roaming charges when they came home."

It had been working hard over the last ten years to fix this "market failure", it said.

"Eliminating roaming charges is one of the greatest and most tangible successes of the EU," the Commission added.

The new legislation means travellers can use their regular data allowance anywhere in the EU.

But consumer organisation Which? warned that exceeding data allowances would still be chargeable.

Exceeding agreed minutes, texts and data would still be charged in the EU as it would in the UK, with providers charging different rates, said Which?

In addition, different providers included different countries in their roaming territories, such as the Channel Islands and Switzerland.

Which? said mobile users should check the detail of their tariffs with their phone providers to avoid being caught out by surprise charges.

Analysis: Rory Cellan-Jones, technology correspondent

Just a few years ago, European mobile phone users travelling across the EU could face a nasty bill when they got home.

Over the ten year period that the European Commission has been bearing down on roaming charges, that risk has receded, and from today customers can call, text or download data on the same basis as they do at home.

But the Roam like at Home legislation only applies to travellers - calling another EU country from home will still incur extra charges.

The mobile phone industry battled long and hard with Commission over the details of the plan to end roaming, warning it could mean higher prices at home.

The Commission saw it as a vital part of creating a digital single market - but also as one of the few examples of an EU policy which would resonate with the public.

Brexit question mark

It would be up to a future UK government to decide whether to have the EU price restrictions on roaming or not after the UK leaves the EU.

They are contained within a European regulation, not a directive, so they have not been incorporated into UK law.