"Come out, come out, wherever you are"

The extraterrestrial has spawned good books, mediocre art and bad movies; provoked serious speculation and a new science called astrobiology; and triggered a 400-year religious and philosophical debate, all without putting in a single appearance. If life exists on Earth - a nondescript planet orbiting an undistinguished star in a neither-here-nor-there galaxy in an ordinary corner of the universe - then it ought to exist on at least some other planets around a proportion of other suns in at least a selection of other galaxies. There are at least 200bn galaxies, and each may be home to 200bn stars. Even if the evolution of a sentient, intelligent, technologically aware civilisation is rare, the firmament should still be fizzing with life.

But, as the physicist Enrico Fermi once asked, in a question now known as Fermi's paradox: "Where is everybody?" The alien is one of two possible answers to life's great question: is all this just for us? But if the alien exists, then alien civilisations would have begun to ask themselves the same question perhaps a billion years ago. The heavens should be ringing with long-distance calls, the galaxy buzzing with randomly directed robot probes. Forget about UFOs, Area 51 and the Men in Black: nobody so far has tried to get in touch.

That hasn't stopped humans hoping for a call from Alpha Centauri or Andromeda Central. A new exhibition - it will open in October - in preparation at the Science Museum in London will underscore just how ardent that hope has been. Contemporary humans have populated the heavens with Little Green Men and Ming the Merciless, Mr Spock, the Klingons and the Borg, Daleks and Monsters from Mars, the Pod People and the Thing from Outer Space but wistful wondering about other worlds began a long time ago: before Copernicus, Galileo and others had firmly established that Earth was a planet, just like Venus or Mars.

Epicurus wrote to Herodotus in 300BC proposing there could be "infinite worlds both like and unlike this world of ours" inhabited by "living creatures and plants and other things we see in this world". Kepler thought it highly probable that Jupiter was inhabited and Christian Wolff in the 18th century even worked out what a Jovian might look like. If bodily size was proportional to the eye, and the square of the diameter of the pupil was inversely proportional to the intensity of available light, and if Jupiter was 26/5 times further from the sun it would get 5/26 times the available light so, bingo, a Jovian would need to be 1,400ft tall.

The astronomers Herschel and Bode proposed that even the sun might be inhabited, Benjamin Franklin wondered about the constitutions of the people who lived on Mercury, so close to the sun; and a Scottish clergyman called Thomas Dick in 1828 calculated there might be 2.4bn inhabited worlds within the visible universe. In 1837, he went further. He reasoned that the population density of England at 280 souls per square mile meant 53 billion lived on Venus and more than 8 trillion people might dwell on the rings of Saturn.

Puzzlingly, until 1898, when HG Wells wrote The War of the Worlds and supposed an invasion of technologically superior, malevolent Martians, human attitudes to citizens of other planets were mostly benign. Emily Bick, curator in South Kensington and one architect of the exhibition on the science and psychology of aliens, reckons that Europeans began to project their fears of the unknown on aliens at about the time they completed their exploration of the globe.

There were unearthly phenomena in ancient texts - Ezekiel's chariot, perhaps, in the Bible - but in general, until the 20th century the Earth itself might conceal sufficient objects of fear. Who needed alien abduction, rectal probes and sperm theft when witches directed your love affairs and fairies stole male babies to fortify fading fairy virility; when the dead could rise from the grave as vampires and siphon off a nightly supply of blood? Who needed the invaders from outer space in the television series V, when monstrous reptiles routinely stalked the night?

"Think about things like Beowulf: Grendel, and Grendel's mother and the dragon were all sort of reptilian and scary and gross; just swampy monsters, cold-blooded, evil, not human, not of a mammalian line," she says. "Before, it was maybe the forest, or strange things that happened in the sky, or weird things that happened in a fairy world. Now we have dismissed all that with rationality, so what's left? Aliens are the one thing we don't know about, so we can project all these same fears, these same stories, onto this new world."

She is a serious science-fiction fan and the exhibition will divide aliens into a number of categories, each of which explores some serious aspect of science or culture. Steven Spielberg's ET, for instance, illustrates the concept of neotony: look like a baby and people will fuss about you. "ET has the proportions of a five-year-old child, he has a huge trunk and huge, huge eyes that probably wouldn't even fit in his skull if they were like human eyes: they would be too large. He acts as a child's playmate, they dress him up in doll's clothes like one of their toys."

There are aliens cute and cuddly, aliens hyperintelligent, aliens sinister, aliens from Roswell and Area 51 and of course, the aliens from Hell. Fear is a big part of the extraterrestrial story. One cornerstone of the show will be the Hans Geiger creature from Alien, the monster that impregnated John Hurt, terrorised Sigourney Weaver and nearly got the spaceship's cat.

"She embodies all of the separate kinds of fears we have about aliens. She is both reptilian and takes over bodies, possesses us and has this monstrous feminine aspect. We have things that can transform and shape shift, we have the Thing. The problem with the Thing - that we are afraid of - is that the Thing is able to take over human bodies and we don't know what we can trust," Bick says.

"Most of the alien films of the 1950s are allegories of the cold war; things like Invasion Of The Bodysnatchers - it's the 'reds under the beds' fear: your neighbours are replaced by aliens."

She is keen on the theme of aliens as tricksters who invade by appearing to be like humans. "I Married A Monster From Outer Space is an interesting example: it's a pod people sort of thing. A woman marries a man who turns out to be a monster from outer space, surprisingly enough, and most of the men in the village turn out to be monsters and the way they find out is that they are not able to reproduce. Their wives don't get pregnant. It is an allegory on closeted gay men, in its own way. They save the day by going to the maternity ward, because all the men there with their wives are obviously red-blooded humans, so they go and rout the aliens and everyone is happy."

Aliens often appear as humans, and not just because it saves a bomb on special effects. Aliens are often like us, and they speak English too. Captain Kirk tended to find beautiful women with beehive hairdos on every farflung planet, but Star Trek was an excuse, using stories as allegory, to explore problems here on Earth. The point is, she says, it's easier to identify with aliens who look like us. You can't really have an alien love story with someone who looks like a giant beetle.

The exhibition will explore robot aliens, emissaries from hypercorporate states, cod-Gnostic struggles to discover true reality (think of the Matrix) and aliens with teeth. "There seems to be a primal fear of aliens with large teeth, aliens that are kind of reptilian. We have the Dracula thing. The Predator is stylised to look sort of reptilian and he has four sets of teeth."

It will open with aliens imagined by humans because that's the only data we have. It will explore some of the scientific substance behind all the science fiction conjecture. It will consider the Drake equation, the famous calculation that plays with the probabilities of habitable planets around stars not too different from the sun, and comes up with an indeterminate but awesome number of possible worlds just in one galaxy. It will contemplate the carbon basis and the universal pressures of natural selection that naturally lead to convergent evolution, in which similar environments tend to produce similar designs even though the starting points might be vastly different. It will look at biological universals likely to pop up on planet Tharg, such as bilateral symmetry.

"We have two hands, two legs, two hemispheres of the brain," says Bick. "Most animals on Earth have bilateral symmetry. There are reasons for this. You have a spare of everything, which is quite useful. You can balance yourself, you have a lot of good things going on. So there is speculation that these structures would replicate themselves with aliens and so there is a possibility that aliens on other planets might be like terrestrials, given terrestrial conditions. We don't know for sure."

We don't know for sure because aliens continue not to visit. One in 100 Americans may believe they have been abducted (cue for a short lecture on sleep paralysis, which manifests itself in pressure on the chest and the experience of being taken away) but if the laws of physics extend beyond Alpha Centauri, then they have not. The distances to the nearest stars are awesome, and the energy costs literally astronomical. There has been a brisk debate about why ET never phoned the Seti Institute, and why signals from Earth might never get through to Cygnus X-1 or a planet in the Pleiades.

One - and some serious planetary scientists and astronomers back this theory - is that we really could be alone: that life itself is rare and intelligent life probably confined to one planet. Not so, say others: the raw materials for life as we know it are being manufactured by exploding stars and carried by icy comets all over the solar system and - since the Copernican principle says there is nothing special about the Earth - by extension, everywhere. Which brings us back to Fermi. Where is everyone? Life must be common, even if communities are light years apart.

That could be all it takes to keep the neighbours from getting the message or putting a call through, say engineers such as Christopher Rose of Rutgers State University, New Jersey, in the journal Nature, and biologists such as Clive Trotman at the University of Otago in New Zealand, who did a similar set of sums in his book The Feathered Onion last year. You can't just broadcast a message saying, "Is anybody out there?" The signal dissipates as the square of the distance. By the time you get to Pluto, it's already vanishingly faint.

So you send an ultra-powerful signal as a focused laser beam. How much energy would that take? How long could you afford to transmit? How many directions must you point the transmitter to cover the whole sky? (The answer to that one is 100,000 trillion). And what chance a citizen of an alien civilisation is tuned in when your one-second message whistles by at the speed of light? The arithmetic, says Trotman, predicts one-way communication with both antennae pointing at each other will happen for one second every 10 billion billion years. Assuming, that is, both civilisations are using the same wavelength. Don't wait up for ET. Use your imagination instead.

Read Seth Shostak's guide to fictional aliens.