Three rules. The first arrives as a letter months after you get home, the second can stop you at the rental desk, the third is usually an extra you have to ask for.
The historic centre of most Italian towns is a Zona a Traffico Limitato: an area where driving is restricted to authorised vehicles, either at all times or during set hours. The Italian Highway Code defines it as an area where vehicle access and circulation are limited to certain hours or certain categories of user and vehicle.
Entry is not controlled by a barrier. It is controlled by a camera that photographs your number plate, which is why nothing happens at the moment you make the mistake.
Under article 7 of the Highway Code, entering a limited traffic zone without authorisation carries an administrative fine — commonly cited in the range of €83 to €332, plus notification costs. Amounts are revised periodically, so treat any figure you read, including this one, as an order of magnitude rather than a quote.
Source: Italian Highway Code (Codice della Strada), article 3(54) for the definition and article 7 for the restriction and penalty.
If your driving licence was issued outside the EU and the European Economic Area — a United States, Canadian, Australian or UK licence, for example — Italian law requires you to carry it together with an International Driving Permit, or an official translation of your licence into Italian.
Article 135 of the Highway Code provides an administrative fine, commonly cited in the range of €400 to €1,600, for driving without one.
The permit is issued in your own country, before you leave. No car rental desk in Italy can produce one for you. Travellers do get turned away at the counter with a paid reservation and no way to fix it that day — and if the car was the spine of the itinerary, the rest of the trip goes with it.
In the United States the permit is issued by AAA. In other countries it is usually the national motoring association. It takes minutes to obtain and is valid for one year.
Source: Italian Highway Code, article 135, on driving with a licence issued by a country outside the EU/EEA. The United States embassy in Italy advises American visitors who intend to drive to obtain an International Driving Permit before leaving the US.
Children under 12 years old or shorter than 1.50 m must travel in an appropriate child restraint or booster seat. This is a legal requirement under article 172 of the Highway Code, not a rental company preference.
Ask for the seat when you book the car, not at the desk. It is almost always a paid extra, and in high season the depot can simply be out of them — which leaves you with a car you are not allowed to drive your child in.
Source: Italian Highway Code, article 172.