Barney and Robin may not start out with a whole lot of romantic potential, but they become one of the most important couples in the series by the end of How I Met Your Mother. In fact, seasons 7 and 8 flash-forwards are to their wedding day, and the entirety of season 9 is about their wedding too. In many ways, this makes this the biggest relationship of the series, and many fans expected them to be endgame.
Despite the importance of their relationship, though, they had quite a few ups and downs (and love triangles involving Ted) along the way. From those first flirtations to actually dating to the wedding day, this is how their relationship breaks down by season. (Because so much of the series takes place in flash forwards, the main storyline is the focus, though major moments in flashforwards are also included).
Seasons 1 & 2 - Just Friends
For the first two seasons of the series, the focus is on Ted and Robin, and Ted and Barney are happily just friends. For most of season two, in fact, Robin and Ted are in a happy relationship, so there's no space for her and Barney to have anything more than a friendship together.
There is one notable moment, though, in 'Zip Zip Zip', where Robin acts as 'wingman' for Barney, and it's clear that they have a whole lot of fun together... so much so that Barney thinks the night will end in sex! Robin quickly shuts him down, though.
Season 3 - Barney Falls For Robin (& A One Night Stand)
By season 3, Robin and Ted have broken up, but they are both still struggling to get over their relationship. Barney, meanwhile, has clearly started to have feelings for Robin. He starts noticing her more, complimenting her more, and when she comes up in conversation, he is noticeably flustered and complimentary.
Then, in 'Sandcastles In The Sand', Barney and Robin are drinking and watching her music video over and over, as he comforts her. This then leads to them sleeping together, for the first time in the series. However, this then causes a huge rift between Ted and Barney, due to Ted and Robin's history, and the two do not reconnect as a result.
Season 4 - Barney's Unrequited Love
By season 4, it's obvious that Barney is head over heels for Robin, but she doesn't feel the same for most of the season. He tells Lily about his feelings himself, though, and Ted later figures it out after Barney becomes incredibly upset about Ted and Robin's new 'friends with benefits' arrangement. (Ted then ends the arrangement as a result.) Over the course of the season, Barney's love for Robin is shown growing, and slowly, Robin starts to have feelings for Barney too, but they are both hesitant to start something at this point. By the end of the season though, they have kissed, and it's clear that they are about to take 'The Leap'.
Season 5 - A Real Couple
In season 5, things get serious between Robin and Barney, and they actually start dating. However, as both of them struggle with commitment and communication, there are clearly some issues here - and Lily pushes them to define their relationship beyond just hanging out and having sex. For a while, things between them are going well, and they make a cute couple, but unfortunately, they just don't work out. Partway through the season, it becomes obvious that they are both miserable in the relationship, that they argue all the time, and that they bring out the worst in each other, and they have an amicable breakup. The rest of the season sees them struggle to deal with the fallout - Robin is hurt to see Barney straight back out there sleeping with as many women as possible, and later, Barney struggles to see Robin in a new relationship with Don.
Season 6 - Flirting Friends
By season 6, things start to balance out for Robin and Barney, although there are definitely still hints that they have feelings for each other. Barney is visibly aware of Robin's feelings, and the two do talk about their relationship more than once over the course of the season, always positively.
Barney reassures her when Robin worries that men don't feel 'needed' with her, and in the end, when Barney asks out Nora, Robin is seen looking sad. They may not be actively back together at this point, but it's definitely hinted that they will be again in the future.
Season 7 - Bad Timing (& The Bride Reveal)
In season 7, things start to shift, as Robin is now the one with unrequited feelings for Barney, in direct contrast to earlier seasons. For a while, it seems as though she will move on, though, as Barnet is with Nora, and Robin herself gets together with Kevin. However, in 'Disaster Averted' and 'Tick Tick Tick', their feelings for each other come out, and they end up cheating on their partners with each other. Barney then breaks up with Nora in order to get back together with Robin, but Robin stays with Kevin, and Barney is heartbroken. Later on, though, when Robin and Kevin call it quits, Barney ends up with Quinn, and a memorable speech from the season is all about relationships needing 'chemistry and timing'. Robin and Barney are clearly failing on the timing, until a flash forward at the end of the finale shows that Robin is Barney's bride.
Season 8 - The Robin
Season 8 does a lot of bouncing back and forth between the main storyline and flash-forwards to Robin and Barney's wedding (and the run up to it), so it's safe to say that they are the focus of the series by this point! However, at the start of this season, Barney is still with Quinn. Eventually, their engagement falls apart, and Robin ends up pursuing Barney, but he pretends to be in a relationship with Robin's hated coworker, Patrice. In the end, of course, it's all revealed to be a big play... the last play that Barney will ever run. It's this season that he proposes, and Robin (surprisingly, to some) accepts. From there, they become a great couple, and seem to have left their major issues behind them.
Season 9 - A Wedding, & A Divorce
Season 9 is all about Robin and Barney's wedding, so it's no surprise that they spend most of the season happily in love. There are some issues though, with ups and downs where they lie to their minister, with Barney's mother disliking Robin, and Robin even briefly gets cold feet and considered running away with Ted! However, in the end, Robin and Barney get married, and Barney vows to always be honest with her. There's a strong fan opinion that this should have been the end of their story, but of course, the controversial ending does give them a final chapter - and in the flash forwards, they are shown getting a divorce. It seems that Robin's work is causing problems, that they don't get to really spend time together, and that they are arguing all the time (again). Despite this, they stay friends afterward, even if they can't be a couple.