Babylon 5, in syndication (s1-4) / TNT (s5)
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Caution: potential spoilers.
Where to begin? There were many alien races called First Ones. There was one which was the very first race in the galaxy, and they were immortal. We will meet one of them in the series, named Lorien. But there were other races of First Ones, chiefly the Vorlons and the Shadows. The Vorlons became all about order, and the Shadows about chaos. But that's oversimplifying. In any event, at some point most First Ones went beyond the rim (i.e., left the galaxy). But some remained behind. And the Vorlons and Shadows would have huge wars with each other, like every 1000 years or so.
As for younger races to emerge in the galaxy, the Centauri Republic was formed around the start of the 13th century. And the Minbari (apparently the oldest non-First Ones) were heavily involved with a Vorlon/Shadow war in the mid-13th century, a thousand years before the series starts. Valen (a Minbari not born of Minbari) appeared near the end of the war, established the Grey Council, and helped defeat the Shadows. He also established the Anla'Shok (Rangers). And um... Vorlons and Shadows both secretly visited many worlds over the years. On some worlds, including Earth, they altered people to create telepaths, apparently. In the mid-18th century, some alien species built a "Great Machine" on planet Epsilon 3, before they died out. There were caretakers left behind to run the machine, which um... could do lots of stuff that'd be important later on.
In the 2080s, the Earth Alliance was formed. In 2113, the Centauri Republic conquered the Narn. The Psi Corps was established on Earth in the mid-22nd century. In the 2150s, the Centauri Republic began to decline in power. Later in the decade they made first contact with Earth. Also, the religion of Foundationism was established. Humans started using jump gates and meeting other races. Around the start of the 23rd century, Mars wanted independence from the Earth Alliance. Around 2209, the Centauri left the Narn homeworld after a long war of attrition. Earth helped the League of Non-Aligned Worlds in the Dilgar War, in the 2230s.
In 2245, there was a disastrous first contact between humans and Minbari, which started the Earth-Minbari War, which nearly ended in the extinction of the human race. The war, as well as the events leading up to it, can be seen in the TV movie In the Beginning (IMDb). Finally, it came down to the Battle of the Line, in 2247, after the Minbari got past all of Earth's defenses. What ships we had remaining got together to defend Earth, and were basically being slaughtered. And then... the Minbari surrendered. Just why they did this was a great mystery, though one which will be revealed later....
In 2248, the Free Mars Movement got underway. The Babylon Project began in 2249. The idea was to set up a space station run by Earth, in neutral space, with ambassadors from all the major races, and lots of people coming and going, all kinds of commerce and stuff... Basically trying to ensure something like the Earth-Minbari war never happened again. The first few stations were destroyed. The fourth went online in 2254, and soon vanished (but we'll find out what happened to it eventually). Babylon 5 went online in 2256, I guess. (It's like, in orbit of Epsilon 3, or near there or something.) We first see the station in the pilot movie, The Gathering (IMDb), which takes place in 2257, one year before the start of the series. It is in this movie that the Vorlons send an ambassador, Kosh, to the station.
The series starts in 2258, about ten years after the Earth-Minbari War. The station's commander is a human named Jeffrey Sinclair (Michael O'Hare), and his first officer is Susan Ivanova. Chief medical officer is Dr. Stephen Franklin. Chief of security is Michael Garibaldi. The Centauri ambassador is Londo Mollari, and the Narn ambassador is G'Kar. (Londo and G'Kar were both involved with Earth some time ago, starting prior to the Earth-Minbari War. They don't get along, for obvious reasons.) The Minbari ambassador is Delenn (Mira Furlan), who was also greatly involved in the war, and a member of the Minbari Grey Council. As I mentioned, the Vorlon ambassador is Kosh. You see relatively little of him, and really only one other Vorlon in the series. But they're all Kosh. Sort of. First Kosh good, second Kosh bad. If the Vorlons ever take an interest in anything the rest of the folks are interested in, you can bet it's more important than you thought. Of course they're often disinterested in things you think are quite important. Most of the ambassadors have attachés who are important to the show. Londo's is Vir Cotto (Stephen Furst), G'Kar's is Na'toth, and Delenn's is Lennier (Bill Mumy). There are also representatives of the League of Non-Aligned Worlds. They're important, but not as much so as the other races.
Well, what to say now? I really don't want to say too much about what specifically goes on in the series. You should know that this whole five year series was more or less planned out from the beginning by creator J. Michael Straczynski, so everything ties together quite brilliantly. You have to see the whole series in proper order to truly appreciate it. I don't think I can really give you a good idea of all that happened. In the course of the series there's a war against the Shadows, and there's a civil war between opposing Earth forces after President Santiago is assassinated and Clark takes over, in league with the Psi Corps as well as the Shadows, although I don't think anyone ever is completely aware of everyone who's on their own side. It's very complicated. Mars also secedes from the Earth Alliance. Um... gosh, so damn many things happen, I can't remember it all.
Sinclair left after the first season, replaced by John Sheridan (Bruce Boxleitner), who would be in charge of B5 for seasons 2-4. Sinclair went to Minbar as an ambassador, but later disappeared (I won't tell you what happened to him, but it's very important). In season five (2262), Babylon 5 is commanded by Elizabeth Lochley (Tracy Scoggins). What else should I say about the series? There are technomages, though they're more important to the spin-off series "Crusade" than to this series. There is the Psi-Corps, which all human telepaths are required to be part of, or else take sleeper drugs that subdue their abilities. Psi-Corps gets involved in Clark's government as well as the Shadow War before that, and there's a lot of evil going on there. There is a Psi-Cop named Bester (Walter Koenig) who shows up now and then to make trouble for our friends on Babylon 5. But there are good telepaths in the Corps, as well. And there are rogue telepaths who want to be free and have nothing to do with the Corps. The main group of these were initially led by Byron. There was a telepath named Talia Winters, who... wasn't what she seemed, but even she didn't know that. There's a telepath named Lyta Alexander, whose abilities were vastly enhanced by the Vorlons. There was a Ranger named Marcus Cole who was quite important. Eventually Zack Allen (Jeff Conaway) became chief of security. There was a guy named Warren Keffer who was the first person from B5 to spot a Shadow ship. And, um... tons of other stuff happens. Tons of characters come and go. I dunno what all to say. I really don't. There was a TV movie called Thirdspace (IMDb), which is set in the middle of the fourth season.
After the series:
A TV movie called The River of Souls (IMDb) was set in 2263, half a year after season 5 ended. Legend of the Rangers: To Live and Die in Starlight (IMDb) takes place in 2265, I believe. It was to potentially serve as the pilot of a new series, but that never materialized. Near the end of 2266, A Call to Arms (IMDb) took place, which served as a pilot for the series Crusade in 2267. A few scenes from an episode of Babylon 5 were set in 2278, which is also when the the framework of "In the Beginning" is set. The B5 episode "Sleeping in the Light" is actually set in 2281. There's also the direct-to-video Babylon 5: The Lost Tales and The Road Home.
TV series: Babylon 5 * Crusade
TV movies: The Gathering * In the Beginning * Thirdspace * River of Souls * A Call to Arms * Legend of the Rangers
DTV: The Lost Tales * The Road Home