Asian cinematography as a definition refers to the cinematography of eastern, south-eastern and southern Asia (Far East cinema) and also western Asia (Near East cinema).
The Far East cinema includes naming only the most important ones, Japan, China, Hong Kong, Taiwan, South Korea (North Korea, ridiculous as it sounds, is classified as one of the European cinematographies), Cambodia, Thailand, Iran, Tadjikistan and India. The Near East, cinema-wise, includes first and foremost the cinematographies of Turkey and Israel.
The history of Asian movie art is very long. Its beginnings go back to the optical experiments in the 10th century when the first successful projections with camera obscura were conducted.
The evolution of Asian cinema has run parallel to the growth of her European sister. Nevertheless, this cinematography was then and is up to this day, independent from both Europe and Hollywood, continually evolving and keeping its own, specific cinema style and language.
Until the 1950’s Asian cinema had been so hermetic that it had hardly existed for audiences from 
the European cultural circle. This is particularly striking when the forties and the fifties were regarded as the “golden age” of the Far East cinema. At the time Japanese cinema was flourishing, giving to the world such filmmakers as Akira Kurosawa, Kenji Mizoguchi, Nagisa Oshima and Japanese movies of the New Wave became the inspiration for western filmmakers. In India, Satyajit Ray made the famous Apu Trilogy (1955-1959), upon which Martin Scorsese, James Ivory, Francis Truffaut and Steven Spielberg patterned their movies. 1946, in China the Kunlun Studio was founded, one of the biggest film companies in the country, in which Spring In a Small Town (1948) was produced, the movie directed by Fei Mu It is still regarded as one of the most important movies in the history of Chinese cinematography. Similarly in South Korea, where there were about 100 movies produced annually during the “golden age”. In 1960, two works: The Housemaid by Kim Ki-Young and Aimless Bullet by Yu Hyun Mok are recognised by the critics as the most remarkable Korean movies of all time.
The most outstanding works of that time, classics of the world cinema are: Rashomon (1950), Seven Samurai (1954) and Throne of Blood (1957), all by Akira Kurosawa (because of Kurosawa Japanese cinema became famous the world over, when Rashomon won the Golden Lion (1951) at the Venice Film Festival); Ugetsu (1952) and The Life of Oharu (1952) by Kenji Mizoguchi; Late Spring (1949) and Tokio Story (1953) by Yasuijro Ozu; Awaara (1951) by Raj Kapoor, the above mentioned Spring In a Small Town (1948) by Fei Mu and Pyaasa (1957) by Guru Dutt.
The nineties witnessed yet another revival of the Asian cinema scene. The movies from the Far East began their triumphant march through the most prominent film festivals and won audience recognition worldwide. 1988, Red Sorghum by Chinese director Zhang Yimou won the Berlin International Film Festival, and in 1989, A City of Sadness by Hou Hsiao-Hsien from Taiwan won the Golden Lion at the Venice Film Festival. In 1997 the Iranian director Abbas Kiarostami got the Golden Palm Award at the Cannes International Film Festival for A Taste of Cherry. Besides them, and in the same breath, one can mention the movies of Chinese Chen Kaige (Farewell My Concubine, 1993) and Zhang Yimou (Raise the Red Lantern, 1991 and House of Flying Daggers, 2004); Hong Kong born director Wong Kar-Wai (Chunking Express, 1994 and Happy Together, 1997); Vietnamese Tran Anh Hung (i.e. Cyclist, 1995); Taiwanese Hou Hsiao-Sien (Flight of the Red Balloon, 2008) and Ang Lee (Crouching Tiger, Hidden Dragon, 2000); Iranians Jafar Panahi (The Circle, 2000) and Abbas Kiarostami (The Wind Will Carry Us, 1999).
Asian cinema is best described by its specific movie genres. Their exoticism finds many believers outside the continent. The most popular are wuxia movies and J-horrors.
Wuxia is an adventurous genre deriving from ancient Chinese literature, in which heroes were flying masters of the sword. Crouching Tiger, Hidden Dragon and House of Flying Daggers are two of the best-known wuxia movies.
Japanese Horror, on the other hand, is an entirely modern genre, though equally popular, its plots are set amongst the crowded Asian metropolis where the menace comes from the haunted apartments and houses (Dark Water by Hideo Nakata and The Grudge by Takashi Shimizu), or casual objects with supernatural features. (The Ring by Nakata) .
Also, specific Indian cinema from Bollywood turned out to the world and became famous, flooding the world with the light-hearted and melodious love stories.
As popular genre as Bollywood or wuxia is Japanese anime, which are animated stories translating the more noble and classic movie genres. As a result, we get anime-dramas, anime-comedies, anime-horrors and all other kinds of anime movies.
Worthy of mention is also the Hong Kong “blood operas”, in which bloody slaughters are presented in an aesthetically sophisticated way. Master of this genre is John Woo, who excelled in his gun fu productions before being sucked into the Hollywood vacuum to create, amongst others; “Face Off” and “Mission Impossible 2”.
The separate phenomenon is Japanese Takeshi „Beat” Kitano, who successfully makes his cinema (i.e. Hana Bi, Brother).
This is only the shortest possible review, but more you get in the separate articles on the cinematographies of different Asian countries.