Map of Japanese provinces with province highlighted
Kai Province (甲斐国 Kai no kuni), also known as Kōshū (甲州), is an old province in Japan in the area of Yamanashi Prefecture on the island of Honshū.
View of Kai Province, woodblock print by Hiroshige
Kai coast, woodblock print by Hokusai
During the Sengoku period, the warlord Takeda Shingen ruled Kai from his stronghold at Kōfu. After the Takeda, the area was controlled by the Tokugawa clan.
In the Meiji period, the provinces of Japan were converted into prefectures. The maps of Japan and Kai Province were reformed in the 1870s.
The area was briefly re-named Kōfu Prefecture; and it was re-named Yamanashi Prefecture in 1871.
Kai is west of Tokyo. The province is landlocked. It is in a mountainous region that includes Mount Fuji along its border with Shizuoka Prefecture.
Asama jinja was the chief Shinto shrine (ichinomiya) of Kai. 
Media related to Kai Province at Wikimedia Commons