How are the different Links and Zeldas related to each other? Is there any romance between then? Did Link just rescue her and then they parted ways? What if they're brother and sister? Post your thoughts and ideas here. I'm going to start from the first Legend of Zelda and go through the series in the order the games were released (which as far as I knoow is not the canonical timeline, but is the best way to organize the games without stirring up trouble).
In the original Legend of Zelda game, I think Link simply went off on his own after defeating Ganon, possibly to train and become stronger. Zelda's only words to him are "Thanks, Link! You're the Hero of Hyrule!," suggesting there isn't any romantic involvement, though the game had so little character interaction that it's hard to tell. If you count the cartoons as canon, however (which I don't), you might believe there's a one-sided love for Zelda on Link's end, and if you like that, then I'm cool with it too.
The Adventures of Link is to me the first game that implies romance between the two. We see Zelda give Link a kiss in the ending, though considering how old this princess is, it's kind of creepy. Nonetheless, it's the first hint of a relationship.
Moving along to
ALttP, I still think there's romance between the two, and I thought this even before the GBA port clarified things a bit. In the SNES version, the dying words of Link's uncle were "Zelda is your...," causing many fans to think the word left out was "sister." The new translation clearly changed it to "destiny." While this doesn't say much on its own, Link's Awakening (which I believe to be the sequel to this game because it was released next and at this point Zelda games came in pairs) hints that he may have romantic feelings for Zelda. The game was of course just a dream, but Marin is stated to strongly resemble Zelda, and romantic interest seems to be explicitly stated between her and Link, suggesting his fantasy of being with Zelda in the real world.
Now for
OoT. This is where things get interesting. Link's mother is never refered to by name, but is dead. So is Zelda's. It's never said Link's mother wouldn't be the queen of Hyrule. Upon first meeting Link, Zelda comments that his name sounds familiar, though she doesn't seem to recognize his face. This leads me to believe that the Link and Zelda in this game are indeed brother and sister. Why they were seperated as babies remains a mystery, but it's poosible that the Shiekah could only keep one baby safe during the Civil War without risk, so the queen was forced to flee with the other. She was caught up in fights and reached Kokiri forest barely alive, placing the prince in the hands...er... branches of the Deku Tree.
In the Oracles games, little information is provided of Zelda, and she only appears in a linked game. She doesn't have much to say before or after she's kidnapped, though she does give Link a kiss at the end, and he sure gets an adverse reaction from it. Like most of the other games, I find this the possible start of a romantic relationship.
Moving along to Wind Waker, I say the game thing again. After all, Link and Tetra sail off to find a new Hyrule together. Even putting in what little role she has in Phantom Hourglass, there's not much more to be said. They're another happy couple.
Okay, Minish cap, and the Four Sword games in general. Zelda is said to be a childhood friend of Link's in all of the games... and just friends if you ask me. Zelda is more interested in either having fun or maintaining the seal on Vaati depending on the title, and there's no kissing or evidence of them spending the rest of their lives together.
In Twilight Princess, there doesn't seem to be any romance implied between the two. Actually, Link is seen fantasizing about Ilia, so that supports it further. Like in the original game, I think Link rescued Zelda from the evil of Zant and Ganon, but then returned to his quiet life in Ordon.
Finally, we have Spirit Tracks. Oddly enough, I find this game gives more character to Zelda than any other in the past, even though they hadn't met before it started. This Zelda is more selfish than her previous incarnations, but nonetheless takes a liking to Link. Whether they stay friends or marry each other isn't clear to me, but this ia a great take on our favorite princess.
I won't discuss Skyward Sworn since it's not out yet and I don't want to speculate based on what's been shown and then be totally wrong. These are just my thoughts on Link and Zelda's relationships throughout the series. Please share yours.