Sheldon and Amy.  Or as we call them, "Shamy."

You could say Sheldon and Amy met for a number of different reasons.  You could say they met because Howard likes playing World of Warcraft.  You could say they met because Sheldon has an aversion to soiled hosiery.  You could say they met because of how against relationships Sheldon is.  You can say they met for all of these reasons.  All that really matters is that they did meet, in a coffee shop in May of 2010.

Sheldon and Amy have arguably the most stable relationship on the show.  Despite Sheldon's insistance that she "is a girl who is a friend but is not my girlfriend," the two are in constant contact via text or Skype, and once, after a disagreement that they decided was worth ending their relationship over, Sheldon became so distraught that he bought twenty five cats. They later made up, Amy's first task as Sheldon's girl/friend once again being to help rid him of the cats.

From the start, they connect due to their shared feelings about relationships, and before they know it they're spending time together, going to Brian Greene lectures to make fun of the audience toward which he aims, spreading gossip to see what type moves the quickest through their group of friends (not surprisingly to the rest of us "Sheldon and I engaged in sexual intercourse" moved rapidly, while "I'm thinking of growing an herb garden" wasn't passed on at all.)

They shared one kiss, almost a year after they met, which was initiated by a drunk Amy while in her apartment.  Sheldon thought the kiss was "fascinating."

The duo is not in a relationship.  Well, if we had that kind of relationship with another person we wouldn't think that we were a couple.  But for the Shamy, this is about as coupley as they can get.  As Mayim Bialik has said, they are "kinda sorta" dating.