A snowman in Japan is called a 雪だるま, which is basically a two-balled figure with a face. They only have the the two balls (no dirty thoughts please) because their form resembles the tumbler doll of Daruma san.
Dharma dolls (達磨=daruma) are hollow and round Japanese wish dolls with no arms or legs, modelled after Bodhidharma. The doll has a face with a moustache and beard, but its eyes only contain the color white. Using black ink, one fills in a single circular eye while thinking of a wish. Should the wish later come true, the second eye is filled in. Because of the low center of gravity, the daruma doll will return to its upright position after being tilted to the side. As such, it has become symbolic for optimism, persistence, and strong determination.
This is MY Dharma doll, that was given to me by a co-worker ages ago. As you can see my wish has been made, but it hasn't yet come true. As soon as it does, you all will be the first to find out!
Right now Kentucky Fried Chicken (referred to as "Kentucky" over here) is giving away plates with the Snowman character popularized in book form by Raymond Briggs. The cartoon version of the story has been a staple of British television viewing every Christmas. Though I was familiar with Briggs' work (Fungus the Bogeyman, When the Wind Blows), I had never seen The Snowman prior to my arrival in Japan. It's a charming piece of animation with a catchy tune, "Walking in the Air" and a Scottish soft drink called Irn Bru has done a spot on parody of the toon.