I haven't forgotten you for a single day. Every day my thoughts... always turn on you, bastard.The person in question is obsessed. Because of some small slight, they stalk the person down, spending all of their time thinking about them. But it's not like they're in love or anything. It's just that this person has to pay for what they've done. This trope comes in a few flavors, either they realize they are in love and drop their revenge, they fail and settle for romance instead, or perhaps they succeed in their revenge and have it end tragically for the other party either having no idea what to do with the rest of their life or realizing too late that what they actually felt was love. Of course, this is not a bad thing. It makes for an interesting premise to have a love story centered around two people who are supposed to be at odds with each other. Compare with Foe Yay, which is similar but without the revenge subplot, and with more emphasis on shipping. See also The Only One Allowed to Defeat You.
Kyoko, Skip Beat!.
- Sasuke from Naruto spends a good portion of his early background pursuing his brother, who murdered his clan. The flashbacks however show that Itachi was a role model that Sasuke idolized, which is part of why he became so fixated with anger, because he felt betrayed. Sasuke is able to awaken his Mangekyo Sharingan upon Itachi's death (which should only awaken when the person closest to you emotionally dies).
- Kyoko from Skip Beat! is practically Trope Codifier. The fact that she can't even distinguish love anymore because her brain is broken is part of this. Ren actually tries to dissuade her from taking revenge, as both he and Sho realize that the more obsessed she is, the closer she gets to Sho.
- Limelda Jorg from Madlax pursues the eponymous protagonist throughout most of the show, at first in an attempt to get even with her after Madlax assassinates the VIP Limelda is assigned to protect right under her nose. By the end of the series, however, Limelda is so obsessed with Madlax, she perceives their firefights as a twisted Dance of Romance of sorts and wants it to perpetuate indefinitely.
Hello, Unknown Troper. You'll need to get known to lend a hand here.