This is rather a touchy subject for me. I have read books on it and studied it and spent quite some time philosophizing over it.
To start, I'd like to say that I've only come up with theories, and these theories have many flaws and could use a lot of work from more professional people than myself. I really don't know if it's possible; there are lots of doubts that it could really work.
The most famous accountings have been of ninja in Japan. These ninja meditated on a religious level and held it in high regard of importance. When people meditate and their breathing slows down, their bodies attracts things called free electrons. A free electron is something that absorbs light and waves with lower frequencies. These things form a coat around the person in meditation, and the light is absorbed. When something reflects light, it is visible. If something doesn't reflect light, our eyes have nothing to see, and it is not visible.
This coat that is around the person is bringing in all of the light but letting it back out, so the person within can see the world outside, but the world outside can't see in. The same thing goes for sound, because sound waves travel at a lower frequency than light. The person within can hear what the world outside is saying, but nobody on the outside can hear the noises coming from within. Ultraviolet light has a frequency faster than normal light, so it can pick up on something veiled under free electrons. Black holes in outer space also absorb light, and scientists use ultraviolet light to pick up on them.
The flaw of this theory is simple. Free electrons only gather around your body when your breathing is slowed down, but the accountings of the ninja in Japan often record them as disappearing in one area and reappearing in another area, often many miles away from the area of the last sighting. To travel would require exertion of breath, which means that breathing would have to speed up. The only way that the body wouldn't repel the electrons while breathing quickly would be if the electrons had some type of law or rule that made them stick around for awhile after the person went into invisibility, and I haven't found that law yet.
There might be something behind this theory, but I'm yet to prove it. If you have any interest in this, you could read the Art of Invisibility by Doctor Haha Lung, who I have to thank for the basis of my theory.