I'm sure everyone is aware that in Japan people bow to each other. This is the proper way of greeting someone, the proper way to apologize, the proper way to make a request. Waving is reserved for very young children.
I've gotten pretty good at bowing over the years and when introduced to someone I bow just like any other Japanese. If I'm feeling especially "foreign" that day or I can tell that the people I'm with are interested in me, I'll sometimes greet people with a handshake but bow is the traditional greeting. Bowing also makes everyone aware of the hierarchy of the people in the room. The person who's head goes down the farthest is lowest on the totem pole as is the person who continually bobs his head in an apologetic way. On the other hand, the person who barely nods their head or bows only once is definitely the head dog. I tend to be a bobber.
Bowing is also the accepted way of greeting people that you run into and if you see someone you know off in the distance you are supposed to bow. The other day I spied one of my students off in the rice field and called his name and waved to him and he very politely stopped in his tracks and made a deep bow to me. Obviously someone has taught him the correct manners when meeting an elder, but it made me feel like the little kid waving frantically to catch some one's attention. Since the neighbors and farmers in my neighborhood know that I'm from the States, I usually wave to them as I pass on my walks with Choco. Often they don't know what to do with this foreign custom and hesitantly raise their hands in a semi-wave. Especially with the older people, they sort of fumble with their wrists and even I can see that they feel very childish waving back at me. I appreciate their attempts to embrace my mannerism.
Bowing goes on in places where you wouldn't expect. Baseball players will line up and bow before and after a game, the first time to pledge fair play, the second time to thank the opposing team for being a worthy opponent. Another situation that I find humorous is that people will bow away while speaking on the telephone. Usually they are apologizing for some reason and they unconsciously bow to the telephone receiver as they talk!
I came across a couple of signs in the city last week that I'm pretty sure you wouldn't find in the States or at least in other Western countries. The first is a sign warning passing traffic that road construction work is being done and the company is apologizing for any inconvenience. The other sign was at the gasoline station and is asking customers to take their trash home with them. Notice the little figure even has taken his hat off to give the proper bow.