So what's the story? More complicated than you think!
for all those who are new to the story, here is a short summary of the many different versions in many different mediums of You Will Live Forever So Long as Your Soul is on the Moon.
The gist of the whole series:the story of two (former) Interpol detectives forced to work for a mysterious private benefactor. While working for this benefactor in a mysterious brownstone apartment, they both discover they can access memories of their past lives in order to solve art crimes (and also learn a few things about each other as well). Widely regarded as both a slow-burn romance through the ages and a (somewhat!) educational show about art history. A more spoiler heavy version can be found here.
The manga:started in 1995 by Senmoto Rie, technically still running, though new parts do not come out consistently, 25 volumes + various artbooks, and other supplemental materials that are considered canon. Volume 10 is where the *big reveal* happens at the end of the French Arc. Output has lagged due to the mangaka's health problems, but is rumored to be ending sometime in 2020 in conjunction with the last animated movie.
The anime: started in 1997 to 2003, detailed the French Arc of the series, up until the big revelation and an epilogue. However, various OVAs and movies have covered the detectives’ other adventures and it was announced in 2017 that there would be a new movie, written with the original mangaka that would be the end of the series.
The k-drama: ran from 2008-2010. Fans are divided on this opinion of this series, because it is less overtly romantic than the original manga, due to censorship. However, the two actors that play the leads are very, very hot. Spans the manga's French Arc but with new characters and side plotlines but also shys away from the big reveal and only alludes to it in its final episodes.
The American television show: From 1999-1999. No one talks about this because it was so bad. Yes, they did make one of them a woman, but that's not the only reason fans hated it. They also tried to make it sort of a "mystery of the week" series and ignored the basic plotline and emotional arc of the manga, resulting in a nonsensical mess.
The HK Film Version: Three movies (2010, 2013, 2016) directed by Saturn Tsai known for her moody, non-linear films shot with saturated cinematography. some fans of the films have no idea that it was originally a manga. In fact the first movie in the series is actually the Hong Kong arc, which, due to when the movies were made is set in the 1980s, with no mention of detectives (however, the reincarnation aspects are alluded to, near the end) and the characters are in rival syndicates, but without knowledge of the series, one would think it was just a tragic romance. The first movie, has significant influence from Wong Kar Wai and Edward Yang, while the second movie is shot more like a 90's-00's McG music video. The director has said that the project itself required somewhat of a chameleon's touch, as exemplified by the final movie in the series, which is a bunch of shorts in many differfent styles.
More comprehensive summaries can be found in the Spoiler Section.