First, here's Rick's page of money tips (it was listed in Edgar's post above, but I wanted to make sure you saw it, as it's excellent). Read all the links, and you'll be an expert: http://www.ricksteves.com/travel-tips/money
Second, there are two limits: what you bank will allow per day, in dollars, and what a local ATM will allow, in local currency (in this case euros). When I was in Portugal in 2002, there was a nation-wide limit of €200 per ATM withdrawal (I've read that this has been changed). In Italy, I've encountered various limits at times, but not recently (on my trip to Sicily in April-May 2014, I withdrew €480 and €530 without difficulty). If you need more than the ATM's limit, and you're still under your bank's daily limit, you can make a second withdrawal (but this may incur extra fees - see below).
On the US bank side, my TD Bank allows $730 per day. Some posts here indicate that this is on the high side (in other words, other US banks have lower limits), so you do need to check. You can usually get your bank to raise their daily limit if you wish.
Third, there are no fees at bank ATM's in Italy. Your bank will most likely add fees. Sometimes it's a percentage of the transaction, sometimes it's a flat fee per transaction, and sometimes it's both (or neither). My Chase account charges $5 plus 3% per foreign withdrawal. My Smith Barney account charges $0 plus 2% per withdrawal. And my TD account charges $0 plus 0% per withdrawal. Guess which one I use? (Of course, I like having back up, but that's what the other accounts are; I would only use them if I couldn't access the TD account for some reason).
One last tip: the Italian ATM will not show dollars at all. It will simply ask you how many euros you want. So, find out your banks' daily withdrawal limit in dollars, and convert this to euros at XE: http://www.xe.com/. This way, when you arrive jet lagged and still thinking in dollars, you'll know how much to take out. I like to use amounts that end in 30 or 80 (like the €480 and €530 above), so that I am guaranteed at least one €10 and one €20 bill. If your first amount is rejected, try a smaller amount.