Heritage Square itself is great and has all different kinds of events going on. It is also literally right by the Arizona Science Center. The setting is great, the plants and buildings are beautiful and unique, and the entire area is a pleasure to be in.
World Festival: My last visit was for WorldFest, a cultural festival celebrating cities from around the world. Most of it was a less-than-thrilling and a lot geared towards children (which I don't have). They did have a massive "Beer & Wine from around the world" stand that was extremely fun. Beer tickets were $5 for 4 (mildly annoying that each ticket was $1.25 instead of just $1) but each ticket could get you a sample or 4 tickets a glass. 4 samples do add up to a glass, so we opted to get all samples and try as many things as possible.
Japanese Festival: My first visit was for the Japanese Festival, which was very fun. There was a ton to see, though a lot of it was food. The performances were interesting and the only downside was the amount of people. It was so crammed that you could hardly walk! I guess a bigger venue would have been better for the turnout, but it was still a good time.