Various types of bears are known in many parts of the world. Here in New Hampshire we've got black bears, and it's not uncommon to spot them in the more rural areas now and again. Every couple years one will wander into our neighborhood, pulling down bird feeders and getting into trash cans for a day or two, before wandering back into the woods when it realizes that no more food is being left around. Up north camping areas have very strict rules about food and trash. While, all in all, black bears are not one of the more aggressive bears, we are taught to respect them and to give them plenty of room.
Bears have played an important role in many different cultures through history - more than I could list here. Some examples include the legendary creation of Korea, where it was the son of a bear turned woman who founded the first Korean kingdom. Bears also play a significant role in the mythology of the Ainu people. They consider bears to be mostly benevolent creatures, and their meat and fur are important staples.
Berserkers, Norse warriors, were often said to wear shirts formed from bear pelts into battle (berkserker is derived from the Old Norse for bear shirt), and were known for fighting in a fierce trance-like fury.
In the folklore of some Native American tribes, bears became associated with dreams because of their winter dormancy.
Bears were one of the sacred animals of the Greek Artemis. The cult of Artemis at Brauron was especially linked with bears. It is said that a gentle she-bear was known to visit Artemis' sanctuary at Brauron. One of the girls at the sanctuary provoked it, and was injured because of it. The girl's brothers killed the bear because of this, and Artemis punished the people with a plague. An oracle told the people that the plague could only be removed if young girls played the role of the bear in the rites preformed at Brauron.
It is one of Artemis' nymphs, Kallisto, who eventually became the constellation Ursa Major (of which the Big Dipper, one of the more widely known constellations, is a part of). Kallisto's son, Arcas, became Ursa Minor.
Bears go into the earth in the winter for a dormant period, linking them
with death - and as they emerge in the spring, resurrection and new
life. Especially the mother bears, who often give birth during these winter months, and emerge in the spring with their new cubs.
Their descent into the earth during the winter months also links them to the shamanic underworld, the dream world, and they are powerful symbols of wisdom, ancestral knowledge, and healing knowledge because of this. Their winter dormancy can also be seen as a metaphor for going into the self and communing with the sub-conscious, with connecting with our intuition and instinct.