From 2 June 2007, foreign nationals who intend to stay in Italy for visits, business, tourism and study periods of less than three months do not need to request a permit to stay (law no. 68 of 28 May 2007 published in the Gazzetta Ufficiale of 1 June 2007, no. 126) more info

In this case, foreign nationals must declare their presence in the country in compliance with the requirements set out in the Ministry of the Interior decree of 26 July 2007 (published in the Gazzetta Ufficale 6 August 2007, no. 181)

In particular:

For those coming from a country which does not apply the Schengen Agreement, this obligation will be met by the border control police making a uniform Schengen stamp on their travel document.

Those coming from countries which apply the Schengen Agreement may declare their presence directly in the Questura (police station), filling in the appropriate form within eight days of entering the country. If staying in hotel/accommodation facilities, this obligation shall, in these circumstances, be attested by the foreign national being issued with a copy of the declaration required by law.

Important  - the period of stay in Italy must not exceed three months or the least period provided for by the entry visa, where required.