On Wednesday, MPs voted by 344 to 254 against a clause that would have required the government to negotiate continuing full membership of the Erasmus programme after Brexit.
The Erasmus scheme is a European Union (EU) programme that helps students study in other countries.
Currently, 53% of UK university students who study abroad do so through the scheme.
In 2017 16,561 UK students participated in Erasmus, while 31,727 EU nationals came to the UK.
Erasmus is also involved in vocational training and work overseas, as well as with teachers who want to work or train abroad.
Liberal Democrat MP Layla Moran, who tabled the new clause to the Withdrawal Agreement Bill, said it should have been a "no-brainer".
"For students, young people, those in training and staff who work in the education sector, the Erasmus scheme has been absolutely incredible," she said.
Lucy Dearlove studied at universities in France and Germany as part of an Erasmus year in 2008. The first person in her family to go to university, she says that she "probably wouldn't have thought it possible" to live or study abroad without the scheme, which came with a "generous stipend".
The defeat of the clause does not necessarily mean the UK will not continue with full membership.
A Department for Education official told BBC News: "The government is committed to continuing the academic relationship between the UK and the EU, including through the next Erasmus... programme if it is in our interests to do so. The vote last night does not change that.
"As we enter negotiations with the EU, we want to ensure that UK and European students can continue to benefit from each other's world-leading education systems."
The UK is scheduled to leave the EU on 31 January, once the withdrawal agreement passes through Parliament and is ratified in the European Parliament.
There will then be a transition period until the end of the year, during which the UK-EU relationship will continue much as it is now - including the Erasmus scheme.
That means funding for programmes in the current academic year will continue as before.
There is a funding round for Erasmus programmes that closes in February and any funding agreed then will be honoured even if the placements take place after the end of the transition period - some Erasmus funding awards are for two years.
Universities UK is advising its members to continue applying for this funding as usual and is calling on the government to commit to continued funding through its Support Study Abroad campaign, whether via Erasmus or a new national scheme.
But a report from the House of Lords EU Committee warned the benefits of the programme would be very difficult to replicate with a national programme, that vocational education and training would stop and that leaving Erasmus would "disproportionately affect people from disadvantaged backgrounds and those with medical needs or disabilities".
The Erasmus programme is run in seven-year cycles and the next one will be from 2021 to 2027.
The European Commission has proposed doubling the funding of Erasmus for the next cycle to €30bn (£26bn).
And while the details have not yet been agreed, there have been suggestions it may become easier for non-EU countries to participate, although clearly they would have to pay to do so.
It is already the case that not all the countries that participate in the programme are EU members.
For example, Turkey, Iceland, Norway and Serbia are all what is called "programme members", which means they participate fully.
But even if the government decides it wants to participate in Erasmus after 2021, it may not be able to negotiate that in time for the start of the cycle, so there could be a period when such programmes are not available for UK participants.
"The Erasmus... programme has delivered and continues to deliver significant benefits to the UK and we need to ensure the positives of the programme are not lost as we move into the next stage," Jane Racz, the director of the Erasmus programme in the UK, told BBC News.
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