1. Project Bluebook - Was a series of military led investigations concerning UFOs and Extraterrestrials started on 1952 and then terminated in the 70's. It had two main goals, first was to determine if UFOs were a threat to national security, and second was to scientifically analyze UFO "data". After compiling a ton of data which was then collectively known as the Cordon Report, the United States Air Force then concluded that no concrete evidence of UFOs had been found, that these so called "UFOs" were not indicative of a more advanced technology at that time, and that there was no evidence that the sightings were indeed of a vehicular function. However, this contradicted an earlier report called Project Sign ( Conducted in1942 also by USAF) which said that these sightings were indeed UFOs.
2. Project Rainbow (The Philadelphia Experiment) - These were the earliest known series of experiments concerning stealth in military vehicles. Conducted by the CIA on 1956, the project aimed towards lowering the radar signature of the U-2 spyplane. This project worked...kinda. But some say that Project Rainbow is a derivative or rather a follow up of the more mysterious and famous Philadelphia experiment. The Philadelphia experiment which was conducted in 1943 at Philadelphia Naval Shipyard and it involved turning/making the USS Navy Destroyer Escort Eldridge invisible or cloaked to enemy soldiers. Instead of just turning invisible, The Eldridge was said to have teleported to Norfolk, Virginia and then back again only in a span of minutes. Crew members reported nausea and some had bouts of madness after the experiment. According to interviews with retired officials involved in the project, it was supposed to be an application of Einstein's Unified Theory.
3. Montauk Project - This classified set of experiments were said to be the continuation of the Philadelphia Experiment and was conducted at Camp Hero on Long Island. The project was spearheaded by a numerous collection of American and German scientists and was said to have very large financial and political support from the government. The project was involved with delving into applicable physic techniques, psychological warfare, mind reading, mind control, and time travel. It was then ended on 1983 when a researcher allegedly manifested a monster which proceeded to destroy the facility.
4. Majestic 12 - The words "Majestic 12" is rumored to be a code name for a secret group of scientists, military officers, govt officials, and researchers whose purpose was to investigate, analyze, and supervise the recovery of a crashed alien spacecraft near New Mexico during the 40's. The group was put together by an executive order from Harry Truman after being made aware of the findings from Project Sign. Numerous documents regarding the group were said to have surface but was then declared by the FBI to be bogus sometime after being dug up and then declassified in the 80s.
5. Project Stargate - Was a code name for numerous sub projects conducted to investigate the existence of psychic phenomena specifically remote viewing (gathering information using the mind from a distance) and to further develop these for military usage. At the height of the project, 22 active military and domestic were providing data to all branches of the US Armed Forces. The project was then later transferred to the CIA to be evaluated but results were then considered to be too vague or general to be used in actual operations. This caused the project to be dissolved, and its 20 million dollar assets absorbed into the CIA. Despite it's apparent fall from grace, some military and government officials say that the study on precognition and other forms of psychic abilities were too earlier at that time and that more developments in research and technology was needed in order to fully utilize these powers.
BONUS: Pain Ray (Not exactly that paranormal but weird) - Directed energy weapon that is designed so that a particular (large) area will be repelled of occupants...in a non-lethal manner anyway. These things work by launching high frequency waves at a target, heating them from a distance, kinda like a microwave in theory. It affects the human body in such a dramatic manner such that water and fat molecules are "excited" making them rub against each other producing enough heat for second degree burns. Apart from that, the skin will continuously increase in temperature as long as it is exposed to the pain ray beam. This particularly bizarre weapon is already fielded and is in use at Afghanistan.