Taiyo no Uta - one of the few terminal sickness movies that I actually liked ( it means a lot because usually I detest them), music is involved
Tada, Kimi Wo Aishiteru - a bit too slow-paced, but I liked the cast and the are some beautiful moments
Temptation of Wolves- I've watched it long ago, but I liked it, it was dramatic, interesting and unusual (well, at least from my perception back then)
He Was Cool - I was obsessed with this movie, when I was starting my drama-addict career, it's a typical hopeless girl - bad boy combo but I loved it ( I don't know if you can find it in good quality though)
Chilling Romance - brilliant and unusual ghost story - romance combination. The male lead is really one of a kind and it's a perfect role for Lee Min Ki
Kimi ni Todoke -extremely innocent and slow-paced high school love story.
Last Quarter of the Moon - The story is good (even if it's not one to die for)it's visually beautiful, with great nostalgic music (I actually learned to play it on piano I liked it so much), great mods and goth fashion. Good cast too.
I'll go against you and recommend something with a bit of comedy:
Project Makeover- well-done, smart and with a message to convey. A really good rom-com.
My Mighty Princess - I don't even know why I like it so much. But it light, funny and romantic at the same time. Unusual outcome too.
If we are speaking about movies about music, I absolutely must mention Nodame Cantabile Final Score I & II - outstanding leads in an outstanding story about being a musician and living for the beauty of music. However you'd absolutely must watch the drama before ( movies are a continuation) and I don't know if you can stand it, because it has lots of anime-like gags and humor (I loved it but if comedy is not your cup of tea it might be hard)
