The raids were backed by the European police agency Europol and coordinated by judicial authorities in Greece, who were working to track down a group suspected of smuggling large numbers of people in the European Union.
Europol said that two of the alleged group members arrested on Wednesday had been living in Sweden, while the other suspects were rounded up in Austria, Greece and the UK, where the man accused of being the group's ringleader was found.
The people arrested were predominantly of Syrian and Greek origin, with some Palestinians also arrested, a statement from Europol said.
According to Europol, the group is believed to have smuggled around 100 migrants a day since 2013, earning around 10 million euros ($11 million) between them.
Migrants – either in Turkey or on their way to Greece – are understood to have contacted the suspects for help securing further transport through Greece and then via the Balkans to other EU member states, including Sweden.
The suspects are also alleged to have organized fake travel documents and housing for some of the migrants. The police agency said it believed the migrants paid the smugglers using money-transfer services or cash.
The suspects used mobile phones, Europol said, but also managed to use social media and other online communication tools while leading a low profile.
Under the EU's Dublin regulation, migrants seeking asylum in Europe are supposed to be registered in the first country they arrive in. But in practice many try to avoid registration and travel onwards to other European countries.
Sweden has long been one of the most popular destinations for migrants, receiving more asylum seekers than any other country, relative to its size.
Up to 190,000 people are expected to claim asylum in Sweden in 2015. But the country recently introduced tougher border checks and announced it would limit the number of residency permits to EU minimum levels.