In the old days (the bad old nineties), there was just the Panorama chain. Today, in very recent years, thanks to foreign investment, there have been constructed giant shopping malls (each called a centro commerciale) around the outskirts of Rome. They really do make life a lot easier with "one stop" shopping where one can find items not found in any other local in Italy.
Here's the three biggest and best:
In EUR (nice, the newest and something right out of Las Vegas):
In Castel Romano between EUR and Anzio (too distant, but bus from end of Metro Line B at Laurentina):
Next to Rome International Airport (my favorite and easy to get to - just take the airport train from Stazione Roma Trastevere and they have a cinema there, too):