Haven't watched any asian cinema in a while. Always liked films like Oldboy, the Vengeance series, Infernal Affairs, Bittersweet Life, the Chaser, Man from Nowhere, Yip Man, Confessions, Battle Royale.
Haven't seen (m)any in about 5 years now. Any newer recommendations.
Would echo Schmidt's recommendation on both I Saw the Devil and The Handmaiden. The director of the former is also known for A Tale of Two Sisters (maybe one of the best and best known Asian horror movies) and The Good, the Bad and the Weird (a slapstick Asian spaghetti western - good fun).
You may also be interested in the films of Bong Joon-ho, who is another huge Korean director.
He's most known for movies like The Host, Memories of Murder, Mother, Snowpiercer and Okja.
If you want some quieter Japanese movies, all films by Hirokazu Kore-eda are worth checking out. I'm particularly fond of After the Storm and Our Little Sister.
Takashi Miike, while being way too prolific for his own good, has also done some really polished stuff. Thirteen Assassins is a terrific homage to Seven Samurai and loads of fun.