The stereotypical flying saucer is just that, a “flying saucer.” Well, maybe there is an actual reason behind this shape. If spacefarers have been traveling since before our time (when our solar system was just a gleam in some stars' eyes), then the shape of spacecraft may be very well established in the universe. These craft may have actually been seen by the natives here on earth when these spacefarers decided to check us out. This would account for its popularity in myth and folklore. So, let's see what happens if we assume this true.
Right off, the saucer shape just happens to lend itself very well to a huge, super-conducting, magnetic coil. Considering the state of their technology, even something like an exotic particle accelerator would probably work great to generate a large magnetic field. This field would be hundreds of times larger than the craft itself, spanning many kilometers in radius. In space, there is a lot of room for this field to stretch out.
What good would such a huge electromagnet do for space travel? Consider that space is not really completely empty. The vacuum is not a true vacuum. It is filled everywhere with tenuous gases, plasma, particles, and ions, even between the galaxies. Hold that thought.
Recently scientists here on earth have learned that a powerful enough magnetic field can be used to manipulate any object, whether it is naturally magnetic or not. A sufficiently powerful magnetic field will force the atoms of any material to subtly change the shape of their electron shells and become somewhat magnetic and responsive to the field. They have levitated non-magnetic objects (even a tiny, living frog!) against the earth's gravity with huge super-conducting magnets. (The strong magnetism did not hurt the frog.)
Now, with such a large and powerful magnetic field, a spacecraft could form a sort of solar sail to capture the interstellar medium. It would only matter that the medium is there, not what it is made of. Even more, by manipulating this field, the medium could be pushed in any direction at great speed. This gives the spacecraft propulsion. The spacecraft becomes a sort of motorboat with a awesomely huge magnetic propellar. Given the great size of the magnetic field, the acceleration on the spacecraft could become quite substantial. These technologies are just starting to be explored by scientists here on earth, but this propulsion field might be well-established technology elsewhere.
Such a magnetic field would require huge amounts of energy to create and manipulate. Where would this craft get such energy? Certainly not from the Energizer Bunny. Consider the famous equation, e=mc2, or that energy is equal to mass times the speed of light squared. What this really means is that if you could convert matter directly into energy, you would get a really huge amount of energy from a really tiny amount of matter. Failing complete conversion, even converting a tiny percentage of the matter would do. Atomic energy and fusion are based on this principle, and yet these energy sources release just a tiny fraction of the energies possible.
The technology involved here has probably solved the matter-to-energy conversion problem. A tiny amount of the interstellar medium would be more than enough to power this craft. Considering that it is traveling through space at such a high rate of speed, it would be no problem to collect a bit of it on the way. The magnetic field could be used as a sort of interstellar ramjet to collect some matter as the spacecraft is traveling. A ramjet is a supersonic jet that uses the force of incoming air to compress and help sustain the reaction that keeps the jet going. The spacecraft would be collecting fuel with this magnetic ramjet as it travels through space. Also consider how fast this fuel would be entering the spacecraft. Allowing these particles to collide with a target would release a lot of energy in itself.
In theory, the energy available to the craft would be limitless. Furthermore, the faster it goes, the more material it can gather, the more energy it can generate, and the larger and more powerful it can make the magnetic field. The only limits would be any imposed by the technology in use. Superconductors fail at a certain level of current. Particle accelerators have a limited capacity. But who knows what other technology is available for creating large propulsion fields? Considering the nearly limitless power of the propulsion field, this craft could get very close to the speed of light.
Now we get into Relativity. As the spacecraft approaches the speed of light, time aboard it slows down. In theory, if it could reach the speed of light, time would come to a standstill. Let's look at what travel would seem like from within the spacecraft, given this strange time effect. Acceleration would seem constant, and the spacecraft would seem to be able to pass the speed of light. A light-year could be crossed in months, hours, or even minutes or seconds. Given the potentially unlimited power of the spacecraft, it could jump across the universe in seemingly no time. There would be no need for suspended animation or systems that can weather long periods of time in space.
You don't get something for nothing in this universe, and you can't break the rules. To compensate for the short time spent on-board, the surrounding universe would seem to speed up. All numbers considered, the spacecraft never breaks the light barrier. When it gets to its destination, years have actually passed in the surrounding universe, even though much less time has passed on-board. If it travelled ten light-years, then at least ten years have elapsed since the start of the trip. This is a small price to pay if you really need to get somewhere without suffering through years in transit.
At such speeds, any particles of matter, large or small, would be deadly to the craft. How would it avoid a collision? How would it detect such objects before it's too late? Well, it already has a large, long-range propulsion field surrounding it. This field would be affected by anything that enters it, and this distortion can be detected. This could be able to be used to detect any invading particles. The propulsion field does double-duty as a sensory field, which can sense and deflect incoming objects.
Such complicated sensing and manipulation of obstacles and the interstellar medium would require tremendous computing power. It would need to process incredibly complex sensory input, and it would have to perform just as complex manipulation on the surrounding environment. The computer in control of the sensory/propulsion field would not be that much different in its capabilities from a living brain. It could really “feel” the space around it and “push” any part that it needed to. It would be aware of any objects nearby, and would be able to take any action necessary. It would move through space as easily as a fish through water.
This ability to control objects around it is probably very developed to the point that the spacecraft could rearrange objects in any manner wished, to any detail necessary. It could performs feats from simultaneously changing the course of millions of microscopic particles to lifting heavy objects off the ground to creating intricate patterns in, oh, say, fields of wheat. (Just for fun, of course. “Wait 'til the natives get a load of this!”)
This magnetic capability could also be used to buffer the craft against inertial effects. Not only could the craft move itself through space, but keep itself from being torn apart from any sudden maneuvers. If the craft were to enter an atmosphere, it would have a very thick medium to push against, and could pull off some fantastic moves. It could probably sense and use the planet's surface and its magnetic field, as well. The craft would have to be protected against the inertial forces involved.
Ok, now that we've gone over these possibilities, let's get back to the current UFO myths. What are the typical aspects of a UFO? There are the strange magnetic phenomena. They can perform seemingly impossible maneuvers. The saucer shape is so common that it seems cliché in sci-fi movies. These craft are noiseless and silent, and they seem to possess the ability to just hang in the air. And who hasn't seen photos of the intricate patterns in crops? The physics are explored here — are these spacecraft exploring out there?