Before the EU referendum countdown began in earnest, campaigners said they weren’t going to let immigration dominate the debate.

But as the 23 June vote draws nearer, politicians are increasingly arguing about the pros and cons of migration from EU countries.

As ever, many politicians and activists are making claims that fail the FactCheck test. Here’s what you need to know…

How many migrants are coming from the EU?

Immigration from the EU was 270,000 last year, but about 85,000 people left, leaving a balance (net migration) of about 184,000 EU citizens.

The numbers from outside Europe are roughly similar: non-EU was slightly higher than EU net migration at 188,000 people (this doesn’t include tourists, just people who say they plan to stay for at least 12 months).

This means that the Conservative Party will not be able to achieve its “ambition” of cutting total net migration to the tens of thousands by cutting EU migration alone.

Even if migration to and from the EU countries was cut to zero overnight, total net migration would be more than 100,000.

Is EU migration rising?

None of these numbers are significantly different from last year, but net migration from EU countries is slightly higher, which means the 2015 figure is the highest full-year number on record.

There is basically nothing that the UK government can do to limit EU migration without breaking the fundamental principle of the free movement of citizens, to which all members states – and some countries who are not full members – sign up.

The back story here is that net migration to Britain was actually negative for most of the 1960s, ’70s and early ’80s – more people left than came.

It was below 100,000 people a year until 1998, then rose steeply as more and more countries in central and eastern Europe joined the EU, gaining the right to live and work in the UK under the free movement rules.

One of the sharpest rises in net migration came in 2004, when the Czech Republic, Estonia, Hungary, Latvia, Lithuania, Poland, Slovakia and Slovenia joined.

How big is the settled migrant population?

So far we’ve been talking about the annual flow of people coming and going, but what do we know about the stable population of migrants living and working in the UK?

The ONS estimated that there were 2,938,000 people with EU nationality living in the UK in 2014. The biggest group – just over 850,000 – were Poles.

If you go by country of birth, rather than nationality, the number of EU-born people was slightly higher at just over 3 million.

The Labour Force Survey says there are 2.15 million EU citizens in employment in the UK.

There is an ongoing row about whether we can trust the official immigration statistics.

Jonathan Portes, Principal Research Fellow at the National Institute of Economic and Social Research, has pointed out that there is a big difference between the official figures and the numbers of migrant workers registering for a national insurance number.

Migration statistics say 739,000 people arrived from other EU countries between 2010 and 2014, but the number registering for a national insurance number was 1,537,000.

The ONS published a report on this earlier this month, saying most of the difference was down to short-term migration – people coming to Britain to do seasonal work, then returning.

But Mr Portes still thinks the official migration number is too low, saying: “I think there clearly are more recent EU migrants present and active in the UK labour market than suggested by the official statistics.  By how much, we don’t know.”

Do EU migrants take British jobs?

It seems a matter of common sense that a big influx of foreign labour will make it harder for native Britons to find work. But several academic studies suggest that’s not necessarily true.

Research commissioned by the Migration Advisory Committee found that “inflows of working-age EU migrants did not have a statistically significant association with native employment” between 1995 and 2010.

This NIESR study from 2012 looked at national insurance numbers and found a “general lack of an aggregate impact of migration (EU and non-EU) on unemployment.”

Jonathan Portes and Sara Lemos looked at the British Labour market in the two years after the EU expanded in 2004.

They concluded: “Despite anecdotal evidence, we found little hard evidence that the inflow of accession migrants contributed to a fall in wages or a rise in claimant unemployment in the UK between 2004 and 2006.”

Does high immigration slash wages?

Again, you would think that more people in the labour market would tend to drive down wages. Again, you may well be wrong.

This paper from the Centre for Research and Analysis of Migration (CReAM) at UCL found that immigration depressed wages slightly for the very low paid between 1997 and 2005. But the overall effect over the whole wage distribution was slightly positive.

A study by the IPPR think-tank reached an opposite conclusion: there was a slight negative effect on wages from 2000 to 2007.

But the authors stressed that the effect was tiny. An increase of 1 percentage point in migrants as a share of the workforce was associated with a decrease in wages of 0.3 per cent.

They said: “Even a large influx of migrant workers is only likely to affect average hourly wages in the UK by a few pence an hour, either upwards or downwards. Compared with the impact of changes to the tax system (for example), the effects of immigration are negligible.”

Neither of the studies was able to separate EU and non-EU migrants.

What about pressure on services?

Another CReAM paper from 2014 looked at whether different kinds of migrants paid in more to the UK in tax than they took out in services like social housing, benefits, healthcare and education.

The authors found that immigrants from the European Economic Area (the EU plus Iceland, Liechtenstein and Norway) made a positive contribution in every year from 1995 to 2011.

“Notable is the strong positive contribution made by immigrants from countries that joined the EU in 2004,” they added.

Earlier this watch the campaign group Migration Watch published a paper that claimed to “extend” the CReAM research while applying it to just one year: 2014/15.

Migration Watch came to a very different conclusion: immigrants, both from the EEA and outside, were actually a drain on the state in that year.

It’s fair to say that you only get this result if you change part of the methodology of the original study, and assume that migrants pay no business taxes.

If you stick with the original method, you still get a slight positive contribution from all EEA immigrants:

In any event, both CReAM and Migration Watch appear to agree that native-born Britons take out far more money from the system than they pay in, compared to any kind of migrant.

The Office for Budget Responsibility accepts the basic point that immigrants tend to improve the country’s finances. A major cut in immigration would mean tax hikes or more spending cuts, the watchdog has said.

This is because migrants tend to be younger and healthier, so they are more likely to be in work and paying taxes and less likely to be retired or to need healthcare.

HMRC recently said that recently arrived EEA nationals paid £3.1bn in income tax and national insurance in 2013/2014. They took out £0.56bn in HMRC benefits.

Of course, this kind of macroeconomic analysis cannot capture the experiences of people living in areas that have seen very high levels of EU immigration, so we can’t rule out incidents of pressure on local services.

Will immigration stop if we vote for Brexit?

Britain will only be able to slow or halt immigration from the rest of the EU if it pulls out of the “free movement” agreement that lets EU nationals live and work anywhere in the bloc.

Even if we vote to Leave on 23 June, this might not happen.

Other countries that have chosen to remain outside the union, like Norway and Iceland, have had to accept free movement as the price of access to the EU’s internal market.

It’s possible that Britain might not follow this option, but every serious recent economic analysis of the costs and benefits of Brexit has said that EEA membership is the least risky option.