The claim: EU citizens in the UK, and UK citizens in the EU, will be able to use their European Health Insurance Cards (EHIC) after the UK leaves the EU.
Reality Check verdict: If you are already living in another EU country on the day the UK leaves the bloc, that is correct, your EHIC will continue to work.
After that date, for EU citizens wishing to travel to the UK or UK citizens wishing to travel to the EU, it is unclear about what will happen because no deal has yet been reached.
At a press conference with the EU's chief negotiator Michel Barnier, the Brexit Secretary David Davis talked about the progress that had been made on citizens' rights in the latest round of Brexit talks.
He said both the EU and the UK had agreed to protect existing EHIC arrangements.
What is the situation now?
The European Health Insurance Card (EHIC) entitles citizens of EU countries to state-provided emergency medical treatment within the EU country they are visiting.
This is separate to the arrangements for UK citizens living in another EU country, who cannot use EHIC in the country they live in but are entitled to the same state-funded healthcare as citizens of that country.
EHIC works in any EU country, as well as Switzerland, and the European Economic Area (EEA) countries of Norway, Liechtenstein and Iceland.
It will continue to work for as long as the UK is in the EU, while Brexit negotiations continue.
According to the current timetable, that is until midnight on 29 March 2019.
The UK has reciprocal health insurance deals with a few non-EU countries too, including Australia and New Zealand, under which visitors can receive free urgent treatment.
These will be unaffected by any forthcoming EHIC changes.
What has been agreed in Brexit talks?
The latest joint EU/UK document on the Brexit talks says that citizens who live in another EU country on the day the UK leaves will be still be eligible for the same healthcare as citizens and will still be able to use the EHIC scheme when visiting another EU country.
This includes citizens who work or study in another country or are retired there.
The agreement has not been reached on whether EHIC would be available to those who travel to, or go to live in, another EU country after the UK has left the bloc.
The Independent has reported that the EU wants discussions on that to be included in the negotiations on the future relationship between the UK and the EU, which will come only after "sufficient progress" has been made on the "divorce" issues.