Elliot and Olivia have been partners since 1999 until he retired in 2011. From the beginning, there was a chemistry that had (and still have) some viewers wanting to get them together, especially during the Stablers' separation. But from the show's view-point that wasn't the story: Elliot and Olivia held onto a 12 years platonic relationship. If they got together, it would've been the end of the show. Elliot and Olivia did get close enough to trust, rely, and have each other's backs- even risking their job(s) to save the other. Dr. Rebecca Hendrix described their relationship as a degree of mutual reliance and emotional dependence that compromise their effectiveness as police officers: they were too close and if Cragen had split them up, he would've lost his two best detectives.
Elliot was the longest relationship that Olivia ever had with a man (in general). Olivia has helped Elliot many times- taking a pregnant Kathy to her Doctor's appointment (but was in a car accident instead, resulting the birth of Eli) and convinced Bernadette to help Kathleen to admit she's sick with Bipolar disorder. Olivia went to see Kathy one night to inform her Elliot's undercover assignment; Kathy was upset that Elliot didn't tell her himself and attempted to leave. Olivia managed to talk Kathy out of it but agreed with her point. In a later episode, Olivia tried to call Elliot on his cell phone but Kathy picked up- she informed that Elliot got off the hook from a trumped up molestation charge.
Elliot shot Jenna Fox and went on leave- Olivia was convinced that he would return. But Elliot did not; he decided to retire and did not tell Olivia. In a deleted scene, Elliot sent Olivia a small version of his badge and a medallion with a message Sempher Fi