uppehållsrätt

The right of EU/EEA citizens and their family members to live in Sweden without a residence permit, as long as they work, study, run a business, have enough money to support themselves, or are family to someone who does.

You do not apply for the right itself. Instead you show that you meet the conditions when you register with Skatteverket, which then enters you in the population register and gives you a personnummer.

After living here legally for a certain time you can get a permanent right of residence, and Migrationsverket can issue documents that show it. Nordic citizens have their own, simpler rules.