Greek version:
The god Zeus hid the nymph Callisto from his wife Hera by changing her into a bear. Her son, Actas, did not know she was now a bear and while hunting one day came across Callisto. To keep Actas from accidentally killing his mother, Zeus placed them together into the sky as the Big and Little Bear (we know them better by the names Big and Little Dipper).
Iroquois Indians:
A long time ago, far away (New York State) some Indian were chasing a bear through the forest. The Indians ran into three giants who attacked and killed all the Indian, except for three. Then, the three survivors and the bear were transported into the sky, where the chase continues to this day. The bear is formed by the four stars in the cup of the dipper, and the three stars in in the handle represent the three Indians.
Zuni Indians:
For most of the year, the great bear guards the western lands from the frozen gods of the north. In the winter, however, the bear goes into hibernation, leaving the land to be ravaged by the frozen breath of the ice gods. Then the bear wakes in spring and drives the frozen gods back to the north, where they belong.
Basque (region of Spain):
Once apon a time in the land of the Basques, a man was robbed of two oxen by two theives. Very angry, the man sent his servant, his housekeeper, and his dog out to chase the thieves. After a while, the impatient man joined the chase. As punishment for his impatience, the man and everyone else in the story were taken up into the sky. The first two stars in the cup of the dipper are the two oxen, the other two stars are the two thieves; in the handle of the dipper are the servant, the housekeeper, and the master, who is the final star. The dog is the faint star, Alcor.
Chinese:
Chinese astronomers called this