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Does my Italian student visa allow me to travel within the Schengen area?
Yes.
Your type D visa allows you to travel to other Schengen countries provided that you spend no more than 90 days in any 180-day period in Schengen countries other than Italy. One source supporting that statement is https://ec.europa.eu/immigration/general-information/already-eu_en:
Going to another EU country during my long-term stay – more than 90 days
When you stay in an EU country for a long stay, usually for more than 90 days, you will generally be [issued] a long-stay visa and/or a residence permit.
If your long-stay visa or residence permit has been issued by a Schengen area country, you can travel to another Schengen area country for 90 days per 180 day period. You must:
- justify the purpose of your stay;
- have sufficient financial resources for your stay and travel back;
- not be considered a threat to public policy, public security or public health.
In practice, you are unlikely to be asked to justify the purpose of your stay or to show sufficient resources. If you travel completely within the Schengen area, you won't generally encounter border guards, and if you travel through a non-Schengen country, the border guards will most likely let you back into the Schengen area without much fuss once they've seen your visa.