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Under the Treaty of Rome residents of any one member state, and that includes the UK, are free to live, study and work in others. Under certain conditions:
- To live and work they need to have a job
- To study they need to gain admission to an accredited university or college
Hundreds of thousands of residents of member states have come here to live and work.
And British people in their hundreds of thousands have gone to live and work in continental Europe. Our economy has benefited from the labour and skills these migrants have brought, both business and the public services.
The only way the UK could limit immigration from other EU member states would be if it opted to trade under WTO rules and free trade agreements. If the EU offered a custom union to the UK like Switzerland it would almost certainly come at the price of free movement of people.