Looking for a book to read

Does anyone know a good romance book about an immortal or immortal?
Like Tuck Everlasting but more mature.
Basically Edward (Cullen) without the need for blood.

I'm looking for something different to read.
So if you know a type of book like that let me know.
Also I'm open to suggestions on other books, like I said I'm looking for different, maybe even weird. Something that's rarely done, not a popular cliche story line.
Maybe even a book where the girl is the strong character who has to chase the male lead or make him fall in love with her.
I don't know, something different.
Every genre is acceptable.