Be it anime, classics or recent movies, what are they? Any recommendations etc?
I didn't get beyond the 60/70s - Akira Kurosawa and Hiroshi Teshigahara.
In the meantine, from that part of the world, there've been quite a few Chinese, Vietnamese, Korean, while the Japanese mainly made cars and electronics.

Two very different genres:
"Ran" (1980) directed by Akira KUROSAWA. It's inspired by the King Lear plot. Don't be put off by the rather static first 20 minutes or so - it builds up and up. (The name translates as "Disorder" or "Chaos".)
"Love Letter" (1995) written anddirected by Shunji IWAI. It sounds like a pretty standard chick-flick, and that's what I expected when I first saw it. But it's engrossing, wry and full of charm, and leaves you feeling both happy and sad at the end. A mix-up concerning her dead fiance's name leads a young woman in Kobe into a complex relationship with another young woman in faraway Hokkaido. They come to understand that their lives are actually far more intimately bound up than either of them would ever have guessed.
Tetsuo - The Iron Man and The Body Hammer by Shinya Tsukamoto,
1989, 1992 respectively.
Yojimbo is brilliant...
Akira is excellent, even though there are massive holes in the plot.
Princess Mononoke
The voyage of Chihihuro
I´m not sure if The Pillow Book would be considered a Japanese movie, but it´s central theme is very Japanese, and I like it.