in the 11th year of meiji, may 14th
Rurouni Kenshin was first published in Shueisha's Weekly Shounen Jump in 1994, and ran until 1999. The series takes place in the Meiji era of Japan, which is the period immediately following the Bakumatsu, the civil war that took place between the Tokugawa shogunate and the imperalist forces. The main character, Himura Kenshin, is the legendary Hitokiri Battousai, an imperialist assassin whose deeds helped bring the Bakumatsu to a close. The series begins in the 11th year of Meiji, or 1878 in the Western calendar. Since the Bakumatsu, Kenshin has become a rurouni, a wanderer who refuses to kill and who wields a sakabatou, a sword with the dull and sharp sides flipped.
The first arc of the series is often called the Tokyo arc, as it is mostly about how Kenshin settles in Tokyo at the Kamiya dojo, and makes friends (and enemies). His past comes back to haunt him several times, and there's one arc where he slips entirely into the Hitokiri Battousai persona and nearly breaks his vow never to kill again. In the manga, the Tokyo arc covers the first six volumes; in the anime, it comprises episodes 1-27, and includes quite a few filler episodes.
The Kyoto arc begins with volume 7 (act 48) in the manga, and episode 28 of the anime. In general, the anime adaptation of the arc is fairly accurate; there are a few minor differences here and there, but overall it's a solid adaptation and either one is well worth the time.
(There are some spoilers here — I do highly recommend checking out the series for yourself if you haven't read or seen it, but unfortunately there's no way to describe it without giving a few things away.)
The Kyoto arc begins with a former Shinsengumi captain (the highest rank of those allied with the shogunate in the Bakumatsu) attacking one of Kenshin's friends at the Kamiya dojo. This culminates in an all-out battle between the two, in which Kenshin loses himself entirely to his Battousai persona, and it's only the arrival of a high-ranking official in the government that stops them. It turns out that the fight was simply a way for them to test Kenshin's skills in order to see if he was up to the task they have for him: his successor as hitokiri during the Bakumatsu, Shishio Makoto, is planning a revolution, and they want Kenshin to go to Kyoto and kill him.
The bulk of the Kyoto arc hinges on that question: will Kenshin break his vow? Can Shishio be reasoned with? Can be defeated without having to kill him? Kenshin's journey to Kyoto answers some of those questions while opening up many more.
All in all, the Kyoto arc is a fantastic piece of storytelling. It's got all the tropes of great shounen manga — Shishio's gathered together a bunch of fighters to serve as henchmen, so of course there are plenty of great fights, and Kenshin's friends aren't about to let him to face Shishio by himself, so they get in on the action too. I've probably watched the Kyoto arc five or six times by now, and read it just as many, and yet every few years I still go back and rewatch/reread the whole thing just because I enjoy it so much.
The fanlisting portion of this site only covers the Kyoto arc in its original incarnation in the anime and manga, but there have been several new adaptations in the years since.
The first was a two-episode OVA in 2011. Unfortunately, it's not a very good adaptation. Granted, some changes were to be expected given the short running time, but they're drastic enough to sour the whole thing. The only real reason to watch the OVA is for comparison purposes.
Finally, two live-action films of the Kyoto arc, Kyoto Inferno and The Legend Ends, were released in 2014, as a follow-up to the 2012 live-action film adapting the Tokyo arc. The second film, Kyoto Inferno, follows plot threads from the first film while adapting the "Great Kyoto Fire" storyline; the third film completely diverges from the source material in a number of ways. I recommend the first two films, but be prepared for huge changes in the third.