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Applying to study in Sweden is relatively straightforward and should be done through the Studera.nu site. You can apply for self-contained courses or full study programmes through the site. Search for first, second and third level courses (undergraduate to PhD-level) and find the ones you want to apply for; you can apply for more than one programme on the same application.
If you are a national of member states of the European Economic Area (EEA) and Switzerland, then you won't need a residence permit to study in Sweden. If not, then you'll have to apply for a residence permit. More information can be found on the Studera.nu site.
Applicants who are not EU/EEA or Swiss will have to pay an application fee of SEK 900. Deadlines for applications vary, so check the studera.nu site, but deadlines for autumn term are normally in mid-January and in mid-August for spring term start dates. Check the specific course code for your chosen courses, either on the application site or on your chosen institution's site, to ensure you're applying for the correct courses.
As soon as your chosen institution accepts you, you'll need to confirm that you can attend and, if you are not an EU/EEA or Swiss student, you'll have to pay your tuition fees and apply for a visa/residence permit.
You will need to show that you've been accepted onto a full-time course and that you can support yourself financially (and have enough for the full year). More information can be found on the Swedish Migration Board's website.