It does but the order of the scenes are different from the manga. So it's kinda complicated. The book of circus book of murder and book of atlantic are in the manga and the manga is much more advance than the anime.
It's hard to tell what chapter is included in the manga since season 1 of anime has the manga and its own episodes.
This link here provides explanation for the season 1 episodes and which chapters you can find from it.
http://switchanimemanga.weebly.com/kuroshitsuji.html
The second is completely original and separated from the manga. (Not related at all but a fun watch).
While the third season (Book of Circus) returns to the manga storyline at the point which the first season of the anime has previously diverged from (i.e. at the end of the Indian Butler Arc: Episode 15). Then, the Book of Murder is from the manga's 5th arc (i think) is a two part ova. AND, the Book of Atlantic is an anime film that adapts the 6th arc (I think??)
Hope that helps. :))
The manga (being the origin of the story) shares its plot with the anime until about half way through the first season, at which point the animators take off with it as they will. However, the newer versions starting with the title "Book of..." follow manga arcs.
It would be best to view it as its own story and start from the beginning. The manga does show some animated events differently from the first chapter so skipping isn't recommended. Also, the artwork is to die for. Who would want to miss that? :D
Enjoy!
Some characters are a little different too.
Kurushitsuji manga... does it continue the Anime? and which ones are continued? I remember that there's three anime of it, I think.