Non-EU citizen studying at university in another EU country
If the student is enrolled in a university degree programme and hold a residence title for study purposes issued by another EU Member State (Schengen Area. The uniform Schengen short-stay visa is currently valid for Austria, Belgium, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Slovakia, Slovenia, pain, Switzerland and Sweden), can enter Italy without a visa, for a maximum of 90 days in a six month period, as long as the student must complete part of the studies at an education institution in another European Country or want to continue the studies, already started in another member State, in Italy.
To do this, the student must take part in a European Union or bilateral exchange programme with the home country or must be admitted to stay for study purposes in a EU member state for at least two years (these requirements are not required if part of the foreign student's programme of study must be compulsory taken in Italy). Besides, the application for the residence permit must include a document, issued by the home university, attesting that the programme of study that will be taken in Italy is complementary to the programme already carried out in your home university.
Anyway, third-country students, holding a residence permit or equivalent document issued by another EU Member State and valid for Italy, have to apply for the residence permit within 8 days of entering Italy to the Police Headquarters (Questura) in the Province they are domiciled in (Potenza or Matera) if the period is longer than three months.