In the mystical world of Shangri-la, demons and humans live side by side, watched over by a parthenon of ancient Chinese gods. But when normally-civilized demons start to go berserk, the gods require the services of Genjou Sanzo - a Buddhist priest with a magical gun, an evil-banishing scripture and a take-no-prisoners attitude. Aided by the ancient monkey god Son Goku, the half-demon Sha Gojyo and the demon exterminator Cho Hakkai, he sets off on a treacherous journey westward, with armies of demons, dark mages and angry gods all standing in his way...
Students Tokitou Minoru and Kubota Makoto are members of the executive committee of the student council at their school, Araiso Private High School. Their duties are to ensure that the school is safe and the students behave themselves. Licensed to fight and intimidate when needed, the duo takes care of any and all of the problems of their high school, along with the other members of the committee. With rash tempers, flirting and violent behavior abounding, Tokitou and Kubota will help keep the peace at Araiso Private High School – or fight trying!
Ok, these two anime have nothing in common except one thing: they are both made by the mangaka Kazuya Minekura. So if you like his style in animation and characters you might want to watch both Saiyuki and Araiso.
While I agree with tetra's recommendation, there is more to it for me. The whole structure of the series are very much in common. The major difference is the setting they take place in.
The characters got a lot in common, the animation style matches, just as a lot of character traits. Both series have the same kind of feel/atmosphere.
Meet Lina Inverse, a mighty sorceress who fights evil in the name of... greed, gluttony and black magic?! When she meets Gourry, a swordsman whose skill with the blade is rivaled only by his stupidity, Zelgadis, a grumpy sorceror who's been turned into a golem, and Rezo, a priest known for his benevolence (with a dark secret), you know her adventures are just beginning!
Both Saiyuki and Slayers are similar in both atmosphere and execution, yet are quite different in story. Each of the series takes place in a different time than our own, though Slayers is more of fantasy whereas Saiyuki is rather supposed to be ancient, yet the same world. That being said, both series contain a lot of comedy (Slayers having more, however) and have engaging stories that are reminiscent of each other in their own ways. Keep in mind, however, that Saiyuki is deeper and less lighthearted than Slayers is.
If you like the comedy of one of these series, you might like the other; just remember that Saiyuki isn't strictly a comedy as Slayers appears to be. Also, if you like the "team of good vs. powers that be" scenario in one, you'll find much of the same in the other.
Both Series have magic and sword fights and epic adventures that spawn several seasons. If you like magic and adventure then you will like both Slayers and Saiyuki. Both series also have comedy has their main fall back. I also found that both series get much better in their next seasons.
The hostility between Earth and Mars has grown, and a war is drawing close. This, however, does not concern James Links, a rough American cargo hauler who wants nothing else but to patch up the relationship between him and his family. But things change when he discovers the highly sought after orbital frame Dolores, quickly causing himself and his two teenage children to be framed for murder and accused of terrorist actions. Follow them on their journey to Mars as they try to clear their names of these crimes they did not commit, with only each other to rely on.
Okajima Rokuro is a small-time salary man who is carrying documents for his company, when the ship he's traveling on is attacked by pirates. Kidnapped, he discovers to his dismay that his employers' main concern is to ensure the documents don't get into the wrong hands, even if it means sending the carrier to the bottom of the sea. Now, with his former life ruined and his kidnappers seeming comparatively friendly, "Rock" decides to join their merry band of mercenaries, and sets out with a new career to the shadier corners of the South China Sea.
Despite the obvious differences in setting and quality (Black Lagoon's animation is far better), the general feeling you get from Saiyuki and Black Lagoon is the same.
Apart from the great action in both series, Saiyuki and Black Lagoon also share a secret emphasis on character development and offhand philosophy. Both shows follow the adventures of a team of misfit anti-heroes whose group relationships become more and more intimately detailed as the stories progress. The shows also share a similar worldview--characters will semi-routinely discuss a somewhat cynical every-man-for-himself, seize-the-day sort of philosophy at more dramatic moments. And yet neither show takes itself too seriously. There’s character-based humor enough to spare in both.
(Oddly enough, both Saiyuki and Black Lagoon also require a sometimes amusing willingness to suspend disbelief. If an ancient monk can ride around in a jeep and use a credit card then a traditional maid can kill hoards of men with an umbrella bazooka. It’s simple logic.)
In the underbelly of the corporate world, a secret series of battles takes place called the Bus Game, whose participants are solicited randomly via letters in the mail. During the games, teams of three attempt to take into their possession a disk filled with corporate secrets; the winners are given increasingly high cash rewards, while the losers get nothing - or worse, they lose their lives. Toki, Kazuo and Nobu make up the "no name" team, and their goal is to win one billion yen each. Each has a reason to need the money and a secret, disturbing past; but with high stakes and mysterious employers, they can only hope to leave the game alive.
Saiyuki and Bus Gamer are both based of manga by Kazuya Minekura, and if you know her work you'll know her characters are very similar from one show to another. The cast of either show very much resembles the other.