I visited this place three times. First one in my first international trip, in 1987, second time with my wife in 1998, and third time in July 2012 with my family and friends. All three times I had the same impact when entered the Piazza and walked towards its center: my heartbeat got crazy, I got very close to crying, and finally, when sat down in the floor in the center of the Piazza, it was pure peace, like we are not in the middle of the crowd, but instead in the middle of a desert field. Impressive. And when the sun goes down behind the buildings, it is amazing and unforgettable.
Don't forget going upstairs to the top of the tower. It is messy, hard, but worthwhile.