After their kiss in the last part of the episode 'There's No Place Like Home' Captain Swan (Killian and Emma) officially became canon after Emma kissed him outside Granny's diner after Hook told her that he traded his ship for her in order to save her family.
In season 4 they start dating and hit huge milestones in their relationship such as Killian revealing to Emma she is his happy ending or the large, more heart wrenching one when Emma finally admits to Killian that she loves him before getting sucked into the vortex of darkness.
Their relationship progresses in season 5 where not only do they exchange the words 'I love you' many times, but Killian also gives Emma Liam's ring showing his devotion towards her. Another adorable Captain Swan moment is when we find out that Killian bought Emma a house that he hoped they would share with Henry. However they don't get to have the 'white picket fence life' they dreamt of as Killian makes the ultimate sacrifice for Emma and makes her kill him in order to save her and her family. Emma wont let her true love stay dead for long though and travels to the underworld to follow in the footsteps of her parents and split her heart in half, proving their true love and bringing Killian back to Storybrooke.
They first meet when Hook is under cover working for Cora. Emma is Hook's enemy.
Hook guides Emma and her team through Neverland and he obviously has a crush on her even telling her so. However, he is constantly rejected but at one point they did kiss (Good Form 3x05) but it was only gratitude although it was extremely passionate. They become friends after they return successfully from neverland. When Emma loses her memories hook tries to see if kissing her will trigger anything but it doesn't. After she gets her memories back they become better friends. Zelena curses hook's lips and tells him to kiss Emma as it will strip her of her powers. Zelena harms hook and tells Emma that the only way to get him back is to kiss him. She does so and it works but she is now powerless. When they are sent back in time they have several intimate scenes including when she has to make out with his old self in order to keep him occupied. After they get back and Emma learns that he sacrificed his ship for her she kisses him of her own will.
In the beginning of this season Emma pulls away from Hook. Hook confronts her outside of Granny's in episode three and they have a very sweet moment ending with a passionate kiss. Hook and Emma are now in a relationship and in the season finale Emma reveals that she loves hook to Regina after hook is 'killed' in an alternate universe, which was never really real. Emma later then is unable to tell hook she loves him but at the end when she sacrifices herself for Regina she finally tells him before becoming the dark one.
When Emma becomes the dark one, she is transported to Camelot, so everyone travels there to save her. They meet up with Arthur, Guinevere and Merlin. Merlin says he can take away Emma's darkness. He fails, and Hook gets hurt badly and Emma turns him into a dark one, also, to save him. She creates a curse to bring everyone back to Storybrooke and she takes away everyone's memories. Soon, though, Zelena gives Hook back his memories and he is extremely angry at Emma. Emma does all she can to stop him from doing bad things but fails. Hook summons all the past dark ones from the Underworld. But at one point, as Nimue, a past dark one, is harming Emma, Hook comes to his senses and sucks all the dark ones back into Excalibur, but he tells Emma she has to kill him with it so all the darkness will be destroyed forever. It was the saddest Captain Swan scene ever. They share a kiss and then Hook dies. Emma goes to see Rumpelstiltskin later and finds out the darkness was not destroyed but transferred back into Rumple. A pissed Emma decides to go to the Underworld to bring him back.
Emma and Hook's love is confirmed to be true when the two try to use the True Love Scale. Emma chooses to save Hook instead of her own heart, making them pass the test.
At the end of the season finale (5x23 "An untold story") Emma decides to say those three little words to Killian with out any battles or death between them, that " I love you" that Killian and all the captain swan fandom had been waiting for.
During the first episode of the season, Emma starts to get visions of what she thinks is her future and she starts to worry about what the future would hold. She starts to ponder her future with hook. After establishing they have true love, Emma and Killian take the next step in their relationship and they move in together as of Episode 3, The other Shoe. In Episode 5, Street Rats, Emma opens up about her visions and tells everyone the truth. When he finds out, Killian tries to do everything in his power to prevent her destiny from being fulfilled, which results in him keeping a secret from Emma. Killian ends up telling Emma the truth and she forgives him because she would have done the same thing. Emma and Regina get stuck in a wish realm in Episode 10 and Killian and David try to get them back to Storybrooke. In Episode 12, Hook confides to Archie that he wants to marry Emma as well as showing him the ring that he plans on giving her but he doesn't think David approves of him. Archie tells Hook to talk to David about it. At the end of the episode, David gives Killian his blessing to marry Emma but a secret could destroy Killian and Emmas' future. In Episode 13, Killian plans n telling Emma the truth but before he could say anything Emma pulls out the ring that Killian planned on giving her and she tells him that there is nothing stopping them. Before he has the chance to tell Emma the truth, he gets down on one knee and asks her to marry him. She says yes. In Episode 14, Hook tries to burn his memories of him killing Davids' Father but Emma catches him in the act. She tells him that he should have told her about what he did and until he start to trust her, they should take a break. Killian goes to Nemo for advice but in doing so, Gideon traps him on the Nautilus with Nemo and his crew. Emma starts to worry about where he disappeared off to so Snow and Regina try to take her mind of Killian. A few episodes later, Emma and Killian are reunited and he proposes again but this time, the right way. In the next episode, Emma is cooking Pancakes in their kitchen and Killian sneaks behind her. He whispers in her ear, 'Something smells delicious' Emma replies to that by saying, ' It's just from a box' and to which he says, ' I'm not talking about the pancakes'. Emma and Killian start kissing and Emma sits him on the edge of the kitchen table. They are about to make out when Snow walks in on them. In Episode 20, Killian and Emma finally say 'I do' on a rooftop in Storybrooke and less than 10 seconds later they break out into song! The final scene shows them holding onto each other whilst the dark curse draws nearer. The curse alters Storybrooke, institutionalizing Emma and leaving only Henry and Gold "awake," while sending Emma's other allies to their home realms. The Final Battle begins over Emma's belief; if she stops believing in the realms of story, they will be consumed by the Darkness and cease to exist. If all the realms are destroyed, Fiona would gain godlike powers and be able to break the Laws of Magic at will. Emma lays down her sword, allowing Gideon to impale her; this releases Emma's light magic and ends the battle. With both Emma and Gold having chosen good. And finally now Emma and Hook can have their long-awaited, happy ending.
In season 7 Emma and Hook, now married, live their happiest moment, despite Emma's son Henry being away. Several years passed and we see Henry, who is about to be captured by Lady Tremaine, he decides to use the bottle, to call Emma, Hook and Regina, so they can help him. Hook comes along with Regina, and the two succeed in saving him, though unbeknownst to them, the bottle had also brought over the Hook from the Wish Realm, who later appears and knocks the original Hook out to assume his place. When Emma arrives and reveals to Henry that she is pregnant, the other Hook feels guilty and decides to switch back, though the real Hook ends up stabbing him in a fight. The other Hook then reveals that he had a daughter who got trapped in a tower by a witch. After hearing this, Hook gets Emma to heal his counterpart. Happily married and with an upcoming baby, Emma and Hook, decide to return to Storybrooke, and live there, their happily ever after.
Since Hook was introduced many 'Oncers' have been shipping Captain Swan due to their witty banter and common innuendos. In season 3 it became apparent that Hook has a crush on Emma causing the CaptainSwan fandom to grow even more due to their sentimental moments and more importantly their first kiss in Neverland even though Emma insisted it was a one time thing. However, the ship had a rivalry with SwanFire, many debated whenever Emma should be with Neal as he was Henrys father though that soon was no longer an option when Neal died. Soon Captain Swan gained even more followers and became the most loved ship in the entire fandom.