There aren't a lot of romances in Avatar: The Last Airbender — besides the two protagonists, there are Zuko and Mai, Sokka and his girlfriends, and that's pretty much it. However, this narrative choice provided the space to develop the relationship between Aang and Katara throughout the series.
It is fascinating how their bond deepens with each win, but far more so when they experience defeat (showing that they will always learn to get over their differences and make peace because they love each other so dearly). The list given here includes a few crucial instances of their relationship, the impact of each taking their time to blossom into their results.
They Fell In Love At First Sight
From the start, it was obvious that these two characters would end up together. From the moment that Aang wearily opens his eyes after being excavated from his iceberg, he has nothing but a strong affection for Katara.
It begins, of course, like all crushes do: The slow burn of pubescent emotion, the plucking up of courage, the longing glances, before it blooms into its ultimate state, but neither of them have eyes for anybody else (well, at least from a romantic perspective).
They Trust Each Other Absolutely
Excluding brief periods of hostility and furtiveness, Katara and Aang share everything with each other, from the ambiguous contents of their day to their worst secrets and shameful thoughts. In fact, while traveling through the maze tunnels under Omashu, they learn that a true commitment is essential in order to survive the place.
Katara hints that they might achieve some success if they kissed, but they soon discover what "Love is brightest in the dark" means when their only source of illumination dissipates.
She Attacks Anyone Who Tries To Hurt Him
When the Fire Nation soldiers holding Yu Dao for Zuko decide to attack Aang, he continues to insist that he wants to meet the Fire Lord regarding the decision to withhold migration from the Fire Colonies.
They refuse to listen, immediately sending Firebending blasts at him, but before the Avatar can make his move, Katara flies into a rage and Waterbends all but one of the (short-term) villains out of their way. Zuko holds her back physically to prevent her from doing more damage, but this only serves to enrage Aang.
The Official Avatar Aang Fan Club
During one of their many visits to Ba Sing Se (after the conclusion of the Hundred Year War), Aang learns to his delight that a group of girls have formed a clique in his honor — the Official Avatar Aang Fan Club. Watching her boyfriend pander to his latest admirers sparks a flame of jealousy within Katara.
It only becomes worse when one of the club's members refers to her as Aang's "first girlfriend." However, she understands that the basis of his joy is in reliving Air Nomad nostalgia; her insecurities disappear and are replaced by a small pang of guilt. Were there such clubs for the earlier Avatars, though?
She Was Widowed Early
Katara is seen as a wizened grandma in the sequel series, but she has clearly not lost her Waterbending touch (even teaching Korra all the knowledge in her possession).
However, her story has had its own mishaps and miseries, for instance, losing her beloved husband at the ripe young age of 66; one can only imagine her dilapidated state of mind during her mourning phase. In any case, Katara finds the will to continue living, at least on behalf of her family and friends; she is 89 years old as of Book Four: Balance.
She Explains Why Ceremonies Are Important
After a strenuous phase of discussions and veiled threats, the conversation about Yu Dao's political autonomy is finally concluded. Instead of enjoying himself at the massive gala thrown by the coalition government, Aang remains focused on a mysterious lady who keeps appearing in front of him at the most random times (and is visible to nobody else).
Katara observes him stress-meditating and gently explains to him that celebrations are a crucial part of life experience, as they help people consolidate their social relationships by reveling in shared victories.
Their Relationship Was One-Sided (For A While)
Watching the Ember Island Players faithlessly recreate Team Avatar's journey triggers Aang for several reasons. His sense of masculinity is affected by the notion that a woman plays his role, which further aggravates his anxieties about the future of his relationship with Katara.
Aang morosely tells her that he wants to be with her after the war is done with, but she hesitates because "now is not the right time." This obviously doesn't refer to a lack of emotion on her part, but rather the terror that her love might not back it back in the end.
Their Children & Grandchildren
Aang and Katara have three children — Bumi, Kya, and Tenzin, the latter two being Waterbenders and Airbenders at birth. The couple adored their kids, but the older ones later express dissatisfaction at how the youngest was favored by their dad. No family is perfect, right?
When Aang goes off on his Avatar duties, it is Katara who remains at home and raises the little ones. Their combined heritage is unlikely to fail in any case, as Tenzin has four kids of his own (but it must be noted Aang never gets to meet any of them before his death).
They Support Their Queer Daughter
Aang and Katara's only daughter, Kya, comes out to them at some point in the story, the backstory of which is revealed in the extended canon of graphic novels. In her discussion with Korra and Asami, a new couple nervous about exposing their non-heteronormative relationship to the world, she tells them that both her parents were widely accepting of her queerness.
In fact, Kya adds that when she divulged this secret to Aang, he displayed "nothing but supportive" feelings towards her. Furthermore, Air Nomad culture openly celebrates all forms of love, regardless of orientation.
Their Sappy, Sappy Love
Although their romance is barely a thing in the animated show, it is actualized in the climax (their kiss in Ba Sing Se being the clincher). It seemingly gets a lot mushier as they adjust to being boyfriend and girlfriend, roles they have never played before in their lives.
Aang and Katara regularly bat their eyelids at each other, employ the corniest possible terms of endearment, and are generally very physical, all in full view of the innocent public. Maybe Sokka was right to be grossed out by their affections after all. Or maybe they're just really cute together. It's all about perspective, anyway.