Project Rainbow was allegedly an experiment conducted upon a small destroyer escort ship during World War II, both in the Philadelphia Naval Yard and at sea; the goal was to make that ship invisible to enemy detection.
The accounts vary as to whether the original idea was to achieve invisibility to enemy radar or whether the prize sought after was more profound: optical invisibility. Either way, it is commonly believed that the mechanism involved was the generation of an incredibly intense magnetic field around the ship, which would cause refraction or bending of light or radar waves around the ship, much like a mirage created by heated air over a road on a summer day.

The reports goes on to say that the experiment was a complete success... except that the ship actually disappeared physically for a time, and then returned. They wanted to 'cloak' the ship from view, but they got de-materialization and teleportation instead... It has been claimed that the Philadelphia Experiment was partly an investigation into how Albert Einstein's 'Unified Field Theory for Gravitation and Electricity' might be used to advantage in the development of electronic camouflage for ships at sea. Einstein allegedly published his Unified Theory around 1925-27 in German, in a Prussian scientific journal, but it was later withdrawn as incomplete.
This research was aimed at using intense electromagnetic fields to mask a ship from incoming projectiles, mainly torpedoes. This was later extended to include a study of creating radar invisibility by a similar field in the air rather than in the water. The story begins in June of 1943, with the U.S.S. Eldridge, DE (Destroyer Escort) 173, being fitted with tons of experimental electronic equipment. This included, according to one source, two massive generators of 75 KVA each, mounted where the forward gun turret would have been, distributing their power through four magnetic coils mounted on the deck. Three RF transmitters (2 megawatt CW each, mounted on the deck), three thousand '6L6' power amplifier tubes (used to drive the field coils of the two generators), special synchronizing and modulation circuits, and a host of other specialized hardware were employed to generate massive electromagnetic fields which, when properly configured, would be able to bend light and radio waves around the ship, thus making it invisible to enemy observers.
The experiment, conducted at the Philadelphia Naval Yard and also at sea, took place on at least one occasion while in full view of the Merchant Marine ship S.S. Andrew Furuseth, and other observation ships. The Andrew Furuseth becomes significant because one of its crewmen is the source of most of the original material making up the PX legend.
Carlos Allende, a.k.a. Carl Allen, wrote a series of strange letters to one Dr. Morris K. Jessup in the 1950's in which he described what he claims to have witnessed: at least one of the several phases of the Philadelphia Experiment.
Author's Addendum Oct 20002:
I have run across the following documented research concerning the official story of the S.S. Andrew Furuseth:

"Supposedly, the crew of the civilian merchant ship SS Andrew Furuseth observed the arrival via teleportation of the Eldridge into the Norfolk area. Andrew Furuseth's movement report cards are in the Tenth Fleet records in the custody of the Modern Military Branch, National Archives and Records Admnistration, (8601 Adelphi Road, College Park, MD 20740-6001), which also has custody of the action reports, war diaries and deck logs of all World War II Navy ships, including Eldridge. The movement report cards list the merchant ship's ports of call, the dates of the visit, and convoy designation, if any. The movement report card shows that Andrew Furuseth left Norfolk with Convoy UGS-15 on 16 August 1943 and arrived at Casablanca on 2 September. The ship left Casablanca on 19 September and arrived off Cape Henry on 4 October. Andrew Furuseth left Norfolk with Convoy UGS-22 on 25 October and arrived at Oran on 12 November. The ship remained in the Mediterranean until it returned with Convoy GUS-25 to Hampton Roads on 17 January 1944. The Archives has a letter from Lieutenant Junior Grade William S. Dodge, USNR, (Ret.), the Master of Andrew Furuseth in 1943, categorically denying that he or his crew observed any unusual event while in Norfolk. Eldridge and Andrew Furuseth were not even in Norfolk at the same time."
At 0900 hours, on July 22nd, 1943, the power to the generators was turned on, and the massive electromagnetic fields started to build up. A greenish fog was seen to slowly envelop the ship, concealing it from view. Then the fog itself is said to have disappeared, taking the Eldridge with it, leaving only undisturbed water where the ship had been anchored only moments before. The elite officers of the Navy and scientists involved gazed in awe at their greatest achievement: the ship and crew were not only radar invisible but invisible to the eye as well.
Everything worked as planned, and about fifteen minutes later they ordered the men to shut down the generators. The greenish fog slowly reappeared, and the Eldridge began to dematerialize as the fog subsided, but it was evident to all that something had gone wrong. When boarded by personnel from shore, the crew above deck were found to be disoriented and nauseous. The Navy removed the crew, and shortly after obtained another. In the end, the Navy decided that they only wanted radar invisibility, and the equipment was altered.
On the 28th of October in 1943, at 17:15, the final test on the Eldridge was performed. The electromagnetic field generators were turned on again, and the Eldridge became near-invisible; only a faint outline of the hull remained visible in the water. Everything was fine for the first few seconds, and then, in a blinding blue flash, the ship completely vanished. Within seconds it reappeared miles away, in Norfolk, Virginia, and was seen for several minutes.
The Eldridge then disappeared from Norfolk as mysteriously as it had arrived, and reappeared back in Philadelphia Naval Yard. This time most of the sailors were violently sick. Some of the crew were simply 'missing' never to return. Some went crazy, but, strangest of all, five men were fused to the metal in the ship's structure. The men that survived were never the same again.
Those that lived were discharged as 'mentally unfit' for duty, regardless of their true condition. So, what had begun as an experiment in electronic camouflage, ended up as an accidental teleportation of an entire ship and crew, to a distant location and back again, all in a matter of minutes. Although the above may seem fantastic, one must remember, that in the 1940s the atomic bomb was also being invented.