It's been two millennia since the Roman Empire held sway, but it's thriving on a dozen fictional planets. Somehow, aliens love to imitate the glory that was Rome. Here are the greatest interplanetary toga parties.
Note: I'm not including alternate history novels where the Roman empire never fell. Instead, I'm only including "Roman Empire in space" stories, where a copy of the Roman empire spans the stars.
Star Trek has probably the most famous alien Roman Empire of them all. In the episode "Bread And Circuses," the Enterprise visits a planet that's modeled on ancient Rome, except with television and machine guns. As far as I can remember, there's never any explanation for why this planet has developed a society that's exactly like Earth's Roman Empire. It just has. And it's awesome. (Update: various commenters have pointed out the Romulans are sort of Romans as well.)
Doctor Who featured an alien Roman empire in its classic comics series "The Iron Legion," drawn by comics legend Dave Gibbons. This version of Rome runs on mechanical soldiers, but it holds a dark secret: flesh-eating demons are secretly running the Empire behind the scenes, and they develop a taste for the Doctor.
Tales Of The Trigan Empire was a comics series that ran from 1965 to 1982, featuring a version of the Roman empire in space. Says Wikipedia: This similarity even extended to Trigan City, the capital being built on five hills, in a similar fashion to the seven hills of Rome. The Trigans flew atmosphere craft. These vessels were like spaceships, but restricted to the atmosphere of the planet of Elekton. The Trigans' clothing was similar to that of the Romans, with many of the populace dressed in Toga-like garments, or in the case of the soldiery, in Roman-style armour.
Isaac Asimov's Foundation series features a Galactic Empire that's very explicitly modeled on the Roman empire in Gibbons' Decline And Fall Of The Roman Empire.
Empire Of The Atom by A.E. Van Vogt tells the story of a distant future Earth, in the year 12,000 A.D., where atomic war has destroyed the Earth. People retain the skills to build spaceships, but lack other basic knowledge, including simple communication. They create a star-spanning futuristic Roman empire.
The Roma Nova novels by R.M. Meluch take place in the Palatine Empire, centered around the planet Roma Nova. Ruled by Caesar Magnus, the Empire is defended by Legion Draconis, plus the U.S.S. Merrimack, led by captain John Farragut and his second in command, the cyborg Augustus.
Futurama almost showed us a Roman empire planet, but didn't quite manage it. In the episode "The Problem With Popplers," Fry complains about having just returned from the Planet of the Moochers, where everybody tried to mooch off him all the time. But in the episode's DVD commentary track, the writers mention that they were originally going to have Fry coming from a planet where it's the Roman empire, except everyone's a centaur. And this planet's emperor would be named Incitatus, which is also the name of the horse that Caligula had made a senator. Sadly, this never actually made it to the screen, even as an off-hand reference.
Planet Hulk, the greatest Hulk storyline of recent years, featured the Hulk crash-landing on the planet Sakar, which is clearly mean to to be ruled by a Roman Empire-esque civilization. They enslave the Hulk and force him to take part in their gladatorial games for their emperor's amusement.
The Tomorrow People. Thanks to commenter MrTim for this one. In the "Rift In Time" storyline, we visit a timeline where the Romans have access to steam-engine technology. In this universe, twentieth century Earth is called "Rome" and is at the center of a huge interstellar Roman empire. (Technically, it's another "Rome never fell" story, but it is star-spanning as well.)
Imperator is an MMO which does include an alternate-history "Rome never fell" scenario, but it's also a star-spanning Roman Empire, so I think it counts. In Imperator, the Roman empire has reigned for thousands of years, and now includes countless planets.
EVE Online, another gaming universe, includes an empire called Amarr, ruled over by a cyborg Emperor and his Five Heirs.
Actium Maximus: War of the Alien Dinosaurs is a Troma movie (of course) in which alien dinosaurs stage a Rome-style gladiator bout, hence the "Actium Maximus" part. Here's a clip showcasing the decadance of the Actium empire:
The Aquilad by Somtow Sucharitkul is, strictly speaking, another alt-history "Rome never fell" story. But it does feature Romans zooming around on flying saucers, so it's also a star-spanning Rome tale.
Less literal-minded examples:
Star Wars sort of makes a stab at a parallel between its Republic-turned-Empire and that of Rome, but it's never really made explicit that I'm aware of.
Babylon 5, likewise, features the Centauri Republic, an empire which some have suggested is modeled on the Roman Empire.
Hunted, a novel by Canadian author James Alan Gardner, takes place in the 25th century, when humans have formed a super-advanced society called the Technocracy, which one character asserts is "lazy and venal, like Imperial Rome at its most decadent." (Of course, science fiction is full of depictions of empires that have gone soft. But it's not worth including them unless they're either modeled on Rome, or make the comparison explicitly.)
Additional reporting by Alasdair Wilkins.