Carter Hall / Hawkman

Earth-One / Post-Crisis
Shiera was next reincarnated in a very strange way. Her brother Speed’s troubled teenage granddaughter Kendra committed suicide, and Kendra’s spirit then entered her body. She only remembered Kendra’s life, and believed herself to be Kendra, having surivived her suicide attempt. Speed, knowing this was his sister, not his granddaughter, trained her to become the “next” Hawkgirl. After joining the new JSA, Kendra learne the truth. Soon after, Carter was resurrected, not reincarnated, as a grown man with full memories of his past lives.
 * 1)  Shiera Hall / Hawkgirl  - In ancient Egypt, Prince Khufu and his bride Shiera were murdered by the sorcerer Hath-Set. But Khuf swore ebfoer they died that he and Shiera would be reincarnated. And so they were. In the Post-Crisis version, in fact, they were reincarnated many times over the centuries, in each lifetime finding each other and falling in love. In the 1940s, they were reincarnated as Carter Hall and Shiera Sanders, who, after realizing their origin, took the pair of wings made of Nth metal (an anicent metal from Khufu’s period of Ancient Egypt that could defy gravity) to become the superheroes Hawkman and Hawkgirl. They were eventually married and had a son, Hector Hall, who also became a superhero. In Post-Crisis continuity, the two continued their superhero careers into the silver age, before being killed in the Zero Hour event.

Shiera still largely thought of herself as Kendra, and would continue to call herself that. She did not remember her past lives like Carter, and, although attracted to him, was not ready to begin a relationship. Carter understood this.

Although partners, the two were still not together in about 2007, when…

2.  Power Girl  and Hawkman began to have something of a romantic entanglement, kissing at least once, while Kendra romanced Roy Harper / Red Arrow.

DC Animated Universe
In the DCAU is is shown that Katar and Carter Hall are different incarnations of the same person (which could certainly be argued in the comics)
 * 1)  Shayera Hol / Hawkwoman  - Carter, an archaeologist, approached Shayera to show her a Thangarian sword from Ancient Egypt, showing that Thanagarians had been on Earth thousands of years ago. He then invited Shayera to dinner to discuss it more. Batman did research on Hall, and found that he had stalked Shayera, showing up at multiple fights of hers with villains. Nonetheless, Shayera went out with him. It was even suggested they spent the night together before, the next day, going to the dig site in Egypt, a tomb, where the sword came from. There Carter revealed that he had, some time ago, touched an object in the tomb, and had remembered a past life. He had seen the memories of a Thangarian man who, with his wife, had crash-landed in ancient Egypt and helped that civilization thrive, before they were poisoned by an enemy. These two bore the names Katar and Chay-Ara Hol. Carter believed the two of them to be the reincarnations of these beings, and revealed himself in-costume as Hawkman. Shayera doubted that the reincarnation aspect, but remained interested in Carter.

Smallville

 * 1)  Shiera Hall /Hawkgirl  - This version of Hawkman and Hawkgirl were members of the JSA in the 1970s. She died years ago at the time of the main story, and he himself later dies in action.

Arrowverse
After Kendra has become engaged to Legends teammate Ray Palmer / The Atom, they encoutner another incarnation of Carter from the future. Helping him regain his memories, Kendra also starts to remember more, and to feel more, for Carter. She breaks off her engagement to Ray and leaves the Legends to have adventures with Carter, although romance does not yet appear to be happening. �
 * 1)  Kendra Sanders / Hawkgirl  - In a loyal but simplified version of the story, Carter maintains his full memoreis of his past lives, while Kendra, whcih is simply the name of this incarnation of Shiera, does not. They become superheroic partners, but not lovers. The two join the time-travelling superhero team the Legends, but Carter is killed shortly after.