In the ripe direction
The persimmon is - or can be - an extremely beautiful fruit. It looks like a tomato, with a green stem on top and skin ranging (usually) from bright orange to deep red. Like the tomato, almost the whole fruit (except the stem) can be eaten, although the skin is usually discarded. Some varieties have large seeds, although the cultivars raised commercially are typically seedless.
There are many varieties but the type you'll see most often in commercial markets is the squat fuyu and the more elongated hachiya, which is slightly pointed on the bottom. All varieties are astringent when unripe but some - like the hachiya - are unbearably so and should be eaten only when fully ripe and soft. The fuyu can be eaten when it's still firm, when it has a slightly crisp texture, although if ripened further, the flesh becomes soft and pulpy. Ripen the fruit by leaving it stem-side down, at room temperature but out of direct sunlight.
Really ripe persimmons have a slippery, jelly-like texture that many people dislike, despite the fruit's sweetness and delicate fragrance. If you like the taste, but dislike the texture, freeze the persimmon for a couple of hours (you can freeze it for longer, but leave it at room temperature about 30 minutes before eating it, so it has time to thaw slightly). The frozen persimmon, scooped from its thin skin, is almost sorbet-like in texture.
Persimmons are available fresh in the winter months, and they're also dried, so the texture becomes chewy.
Because the persimmon tree is so prolific, many recipes have been developed to use up excess fruit; it's made into cookies, sweet breads and puddings.