The bright pinkish-orange fruit of the Japanese persimmon (Diospyros kaki), also now called kaki in English, is highly prized world-wide for its tangy pulp ... Waiting for persimmons to ripen is a classic test of patience in Japan.
From Haiku World by William J. Higginson
I've never heard kaki used in the US, as Higgonson claims, but then, I've been out of the US for a few years now.
In autumn, right about now, people pick kaki before they can fall and hang them up to ripen. The birds may get a few, but the remaining fruit are quite tasty.