I'll try to keep this brief.
No, really, I'll try.
Stop laughing, guys!
The Retrieval Artist series, by Kristine Katherine Rusch. In Earth's future, we've made contact and trade agreements with various alien races. Unfortunately, humans also sometimes make mistakes involving alien culture, some of which carry the death penalty (or worse). In the cases where innocent mistakes mean alien reprisals, humans can be made to Disappear -- receive completely new and unrelated identities on far-off worlds. Sometimes, the charges against the Disappeared are dropped, or these people need to be found for other good reasons, which is why there are Retrieval Artists, who track down Disappeared. Miles Flint, the main character, starts out as a police officer on the Moon, but eventually moves into a career as a Retrieval Artist. He has the job of both deciding whether the people who want to find the Disappeared are on the up-and-up, and if so, how to find someone who has left no traces behind. Most of the novels double as mystery as well as science fiction, and there is some exploration of alien cultures, especially in the fourth book
Buried Deep. There are six books in the series:
The Disappeared,
Extremes,
Consequences,
Buried Deep,
Paloma,
Recovery Man.
Detective Inspector Chen series, by Liz Williams. I'm just starting the second book now, but I liked the first,
Snake Agent. It's set in something of a future Earth, but includes a lot of Chinese mythology. So Heaven and Hell not only exist, but officials can visit both places. In the first novel, Inspector Chen Wei has to solve a murder and ends up kind of partnered with a demon named Zhu Irzh, who's actually pretty principled for a guy who works in the Vice Squad (promoting vice, not combatting it). Various ghosts, goddesses, and other supernatural beings are involved at various levels. The second book,
The Demon and the City, has just been released in mass market paperback.
Sector General novels, by James A. White. White wrote novels and short stories in the series about an interstellar hospital with with human and alien staff members. One of his most frequently used themes was of various races or individuals trying to overcome evolved survival traits that had been useful in the past, but that got in the way of forming or maintaining a civilization or on-going contact with other races. Many of his early works have recently been re-released in
Beginning Operations and
Alien Emergencies. Since the series was started in the 1960s, some of the stuff is a bit dated, but overall the stories are enjoyable.
I've got more, but my reading journals are mostly at home, so I'll have to wait until I can get to them.