I was wondering if anyone know any drama with robots that seems to have a human heart, and tend to sway the conceptions of what is human.

I came across this and watched it, and it was beautiful.

http://mydramalist.com/1653-air-doll

I have seen anime titles such as Plastic Memories, Chobits, Time of Eve, and I tend to really like these sort of stories.

Thanks in advance :)))

Japan:

Q10

Zettai Kareshi

Taiwan:


Absolute Boyfriend (Taiwanese version of Zettai Kereshi)

None of those dramas are exactly 'deep', though. Q10 is probably moreso, though, than the Zettai Kareshi versions.
Bong Soon - a Cyborg in Love
Ando Lloyd - A.I. Knows LOVE? (haven't watched it yet...)

This one's a movie, but since there's not many robot/cyborg/android dramas, I'll include it in case you find yourself in the mood for a movie instead:
Cyborg Girl (I've seen this one years ago, and loved it)