To apply for a Sweden residence permit, start by visiting the Swedish Migration Agency’s official website to complete the online application form. Prepare all required documents, including a valid passport, proof of financial support, health insurance, and any specific documents related to your purpose (e.g., employment contract or university acceptance letter). Pay the application fee and book an appointment for biometrics at a Swedish embassy or consulate if required. Submit your application before entering Sweden if possible, and wait for the decision, which typically takes several months.