While in European-influenced cultures, the dragon has aggressive, warlike connotations...
In pre-Christian era, dragons were revered the world over as the symbol of good and wisdom.
In China, for example, the dragon has a long history as a potent symbol of auspicious power, both in folklore and art. It represents the embodiment of the concept of the yang and is associated with the weather and water as the bringer of rain. In the old days of Imperial China, it also denoted monarchism and the Dragon Throne. China has nine classical types of dragons.
They are known as:
Celestial dragon: 天龍 (tiānlóng) 
Spiritual dragon: 神龍 (shénlóng)
Dragon of Hidden Treasures: 伏藏龍 (fúcánglóng)
Underground dragon: 地龍 (dìlóng)
Winged dragon: 飛龍 (fēilóng)
Horned dragon: 虯龍 (qíulóng)
Coiling dragon: 蟠龍 (pánlóng)
Yellow dragon: 黃龍 (huánglóng)
...each with their particular definition in ancient Chinese belief. 
 Historically, the dragon was the symbol of the Emperor of China.
In Japan, the dragons are the protectors of the Japanese zodiac and they rule under water. However, Japanese folklore states that they can also fly, that they rule both clouds and storms and that the breath of a dragon causes lightning and rain. Japan has nine classical types of dragons.
They are known as:
O Goncho dragon
Yamata-no-orochi dragon (八岐の大蛇) or Orochi, translated as the Eight-Forked Serpent 
Susanoo slaying the Yamata no Orochi, by Kuniteru
For more information on peculiarities in the depiction of the dragon in various Asian cultures, see:
Bhutan: Druk, the Thunder Dragon, http://en.wikipedia.org/wiki/Druk
Nepal: dragon depicted with Bahirav, http://en.wikipedia.org/w/index.php?title=Bahirav&action=edit&redlink=1
India: Nāga, a Hindu or Buddhist deity often depicted as a king cobra, http://en.wikipedia.org/wiki/Nāga
South India, Tamil: Yaazhi, a mythical creature