It's the civil registry kept by every comune. It tracks who lives where, births, deaths, marriages, and family households. Pretty much every official certificate about your civil status comes out of the anagrafe.
Without being registered in the anagrafe at your address, you don't have residenza, can't get an Italian ID card, and can't access many local services. It's the legal backbone of your civil life in Italy.
"On his first trip to the comune of Foggia, Adnan asked the anagrafe office for a certificato di residenza. The clerk pulled his file and printed it in two minutes."
There's a difference between anagrafe (civil registry) and stato civile (the records of births, marriages, and deaths). Both sit inside the comune, but they're different counters.
Official source
Ministero dell'Interno
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