You want their entire history?
Abby asked Luka out in S7 but on that date they were attacked by a mugger who Luka killed. Cue much anguish on Luka's part and an inability to relly connect with Abby who turned instead to Carter who she was surprised to find was nursing a crush on her. Luka and Abby broke up and Abby was apparently waiting for Carter to make his move, but he said she wasn't over Luka and nothing happened.
Luka has ill advised fling with Neecole, an annoying French waitress played by the otherwise very good Julie Delpy, who was pregnant, or no, she wans't or maybe she was ... Abby was a bit too concerned about this for an ex, hence Carter's turning her down.
Abby was attacked by her neighbour for interfereing in his violent abuse of his wife and ended up staying with Luka (who tracked down the husband and beat the tar out of him). Abby by now was drinking again, leading to a not very edifying confrontation by Carter of Luka wherein he accused Luka of liking his women vulnerable and revealed what Abby had clearly bever told Luka, that she was a drunk.
Carter in the mean time had taken up with Susan in a scarily passionless relationship. There were a number of hints that the writers were inching their way back to Luka/Abby but instead they went carby.
Abby was still drinking (which should have prevented Carter, himself a recovering addict, from taking up with her0 and this was a cause of friction between her and Carter throughout. It's too painful for me to rehearse the thing that was carby but they went belly up when Eric's disappearance (and reappearace) coincided with Carter's Grandmother's death. Effectively they weren't able to be there for each other. Shortly thereafter Carter decamped to the COngo to join Luka who, after a season of sexual acting out and self loathing had gone there to work with MSF.
I should add that Carter came close to proposing to Abby but thought better of it. But she knew ...
Congo, bad tempered men with guns, heroic saves, hot sweaty, stubbly Luka ... give me e moment ... that's better; Carter returns to the US alone. Abby wants nothing to do with him but as they're talking word comes that Luka is dead. Carter goes to retrieve the body but finds Luka alive and sends him back to Chicago with Gillian, a nurse who has been, erm, keeping Luka company under the mosquito nets. Carter sends a letter breaking up with Abby. Nice. Abby v. pleased to see Luka and he her.
S10, the puzzling thing that was Sam and Luka, Abby dates Jake and Carter shows up with pregnant gfriend Kem. Carter and Abby have precious little to do with each other. Cartus is stillborn, Kem goes back to France, Carter has a bit if a fling with Wendall the skinny Social Worker before upping sticks and going off to the even skinnier Kem who he married.
Luka and Sam are put out of their misery and Abby is supportive of him in his gloom. They are clearly good friends by now and after a difficult case Abby does the unthinkable and actually breaks down and cries in front of Luka. Kissing and other stuff follows but they decide, in mutual wrongheadedness, to just stay friends. This lasts all of two weeks when, after Neela and Gallanat's wedding, they pretty much jump one another and it's game on again.
Abby gets pregnant, looks like she might have a termination, says she can't do it and wants them to have the baby. Luka asks her (clumsily) to marry him but she stalls and then, following a shootout at the OK Cor ... erm, the ER involving Sam's deadbeat ex she delivers baby Joe prematurely and has to have a hystorectomy. Luka sobs heartbreakingly.
Joe's OK and following Luka's near thing with Ames Abby changes her mind and asks for another proposal. Realising that every woman really wants her bloke on his knees Luka obliges.
They were married in "I Don't" but before they could leave for the honeymoon Luka got a call from Croatia. Seems his father is very ill and he left in "The Honeymoon's Over" expecting to be gone foe a couple of weeks.
As for what happens next, check the S14 spoiler thread
"Still hot, baby".
