Not many may know that substantial discounts or even free passes are available to AIRE registered citizens when visiting Italy.
For example, all state museums are free at least until 2023, as long as you can show some evidence that you are AIRE registered (like your identity card).
If you would like to check the appropriate ministerial decree can find this information here
And in any event, it may not be a bad idea to print out the circular in question, taking it with you when visiting a museum (together with an ID) to ensure that you will be issued a free ticket.
Do bear in mind that not all museums and palaces are part of the “Beni Culturali” of the Italian state; many are foundations, belong to regions and provinces and so on. Some may offer discounts, but most will ask for the full entry fee, regardless of residence. Other discount cards for AIRE registered citizens may also be available from time to time, and it is worth checking online as this information is seldom publicised.