A valid visa and passport are essential travel documents. While British, Irish and EU citizens need only a passport (and are allowed to stay as long as they please), citizens of the US, New Zealand, Australia, Canada and South Africa need a visa only if they are planning to stay longer than 3 months. If your stay is longer than 3 months you should apply for a residence permit or a permesso di soggiorno. all other nationals need a Schengen visa.
You need not go through customs if you arrive from a EU country. There are no restrictions on import and export of currency.
