Sheldon and Amy or “Shamy” [“ShAmy”] first met when an online dating site paired them up as the perfect match for each other (courtesy of Howard and Raj, who secretly registered Sheldon on a dating site).
Three years later they professed their love for each other. Howard first dubbed them as “Shamy”. Amy more than once by season 7 has suggested that they could move in together. After one such request, Sheldon even admitted to Stuart that Amy loves him.
In season 8 they have admitted that they love each other and were planning a future together until Sheldon burned Amy out and she broke up with him. They reconciled in “The Earworm Reverberation” and consummated their relationship in “The Opening Night Excitation”.
In “The Long Distance Dissonance”, Sheldon drops everything and runs off to Princeton and proposes to her. They got engaged in the Season 11 Premiere The Proposal Proposal.
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