Romance in anime is about as much of a mixed bag as romance in any other medium, with your usual grab bag of repeated elements, along with some very anime-esque spins mixed in for that extra flavour.
Romance is one of the most deceptively hard genres to do correctly. This is because you have to convince your audience that not only are the two characters you've presented to us good characters in their own right, but that they have enough in common to want to date each other, with all the emotional baggage such a decision entails.
It's VERY easy to screw up, especially in anime. Anime romances, whether in the genre of romance itself or just a sub-plot in a larger story, range from believable to flat out hokey. Not helped by flowery melodrama often being anime's default setting.
But sometimes, it really, REALLY works. And when that happens you get a romance that's just as compelling if not more so than the larger story surrounding it. Even decades after their initial release, these ten anime romances still send our hearts soaring.