This is based on the information provided in David Grusch and his witnesses, organized here in a more organic and consistent way.

UFOs have been recovered by the DoD at least since 1933 (Mussolini's UFO). Depending on the source, nine to fifteen UFOs have been recovered. Not all sources are aware of all craft, so the highest number is the most likely. The recovery happened because the UFOs were caught, because they landed (and the occupants left the craft unoccupied) or because they crashed. About half of the craft are in good shape, the other half broke into pieces when they crashed.  

Although ninety years ago things might have been different, currently it's the CIA that coordinates the recovery and storage of UFOs, it's the portfolio manager of the crash retrieval operation. It has a system in place that can discern UFOs while they're cloaked, and if they land or crash, special military units are sent to try to salvage the wreckage. The OGA (Office of Global Access, a wing of the CIA) provides worldwide collection capability, it allows the military to secretly access areas around the world where they would usually be denied. The Air Force Special Operations Command's 24th Special Tactics Squadron, based at Pope Field Army Airbase in North Carolina, has also been involved in securing areas for UFO crash retrievals. OGA and 24th get the UFO into custody and protect the secrecy of it, the actual physical retrieval is by SEAL teams or Delta Force under the Pentagon's JSOC (Joint Special Operations Command).

Retrieved UFOs can't be kept under military control, because they have to keep too many records, so the CIA hands the material over to private contractors for analysis, where it's not subject to government audits and can be shielded with protections for trade secrets. The Department of Energy national labs are the contractors that handle the radioactive material, the aerospace-defense industry are the contractors that handle the non-radioactive material and intact craft. The contractors move the craft around to different facilities, both military bases and contractor facilities, for research. The locations include Area 51 and the Nellis Air Force Base, and the contractors include Lockheed Martin. Due to compartmentalization - and the material being often moved from one location to another - by the time the material gets studied, the person has no idea of what he's studying.

Origins and intentions of the craft's occupants are unknown. It's also not clear if there has been contact and interaction with living aliens. "Biologics" have been recovered, which to the best of my knowledge means limbs, entrails or similar. The people who studied these biologics only have "some idea, not a complete picture" (Grusch) of what they are dealing with, given the difference with human physiology.

Russia and China also have crash retrieval and reverse engineering programs.
With regard to the U.S., one witness said: "As far as I know, we are not able to operate them. There are people who say we have reverse-engineered them and are flying them. I never found any support for that. And found a lot of support for saying we can't figure it out". Another one said: "The U.S. has been able to fly at least one of the retrieved craft", and a different article mentions the "successful reverse engineering of a triangle-shaped craft with unconventional propulsion". It should be noted that there's a difference between being able to fly them and being able to reverse engineer them. The tech is very cutting-edge, and the craft must be moved to places like Italy, Belgium, and Indonesia to do flight testing. 


A witness said he's aware of "at least four morphologies, different structures". For example, an M&M's candy, a saucer, a disk, Bob Lazar's "sport model", all have the same morphology (basic shape) but different structures.

Mussolini UFO: The sources for this are Italian ufology and one of Grusch's witnesses. According to ufology: In 1933, a UFO crashed in Northern Italy (Magenta). It was bell-shaped. It had two passengers who died in the accident. The passengers were 1.80 m tall, blond, with blue eyes and pale skin. Mussolini thought they were German. Guglielmo Marconi (leading the special research office "Gabinetto RS/33" as president of the Regia Accademia d'Italia) and others, who studied the UFO, thought they were aliens. In 1944-1945, with the end of WW2 and the fall of Mussolini, the UFO was given to the U.S. through the help of the Vatican. Grusch says that a witness told him that the oldest alien craft they have is the well-known Mussolini UFO. Grusch also says that no bodies came to the U.S. with the UFO, and that the bell shape is due to the object having lost in the crash the "typical" flat ring that surrounds the centre.    

Lockheed Martin UFO: The sources for this are Sen. Harry Reid and Grusch's report of his talks with Reid. As described by Grusch, this UFO consists of "bits and pieces" recovered in the 1950s. The exact facility's address is in Grusch's written report.
In 1993 Ben Rich, Director of LM's Skunk Works who died in 1995, gave a presentation at Wright-Patterson Air Force Base and UCLA. He reportedly said: "The U.S. Air Force has just given us a contract to take E.T. back home"; "We already have the means to travel among the stars, but these technologies are locked up in black projects and it would take an act of God to ever get them out to benefit humanity. Anything you can imagine, we already know how to do"; "It is time to end all secrecy on this, as it no longer poses a national security threat, and to make the technology available for use in the private sector".
As reported by Reid and then Grusch: LM wanted to get rid of the material because, given the level of secrecy, they can't call anyone to do the reverse engineering, and therefore having the material becomes useless. Reid founded the AAWSAP to create a suitable holding structure, and then tried to obtain, manage and study the aforementioned material, but the CIA did not allow it.

Nellis UFO: The sources for this are Martin J. Powell, who did a great job at studying the sighting, and one of Grusch's witnesses.
In 1994 a UFO was filmed while flying over the Nellis Air Force Base, Nevada. The Air Force surveillance cameras were at two locations, known as S-13 and S-30. The videotapes were sold to TV producers by a former Air Force employee who worked on the range. In 1995 the American TV shows Hard Copy and Sightings broadcast the videos. The UFO doesn't make spectacular manoeuvres, it just floats, almost like a balloon, and it's mostly out of focus.
In 2002 Powell wrote an extensive report: Prologue, Location, Appearance, Performance, S-30 Transcript. Most importantly, he sketched what he thought the UFO seemed to look like: image.
The story would end there, with what could have been a balloon or other debris, if it weren't for one of Grusch's witnesses, who many years later describes one of the craft he saw: it "looked like a chopped up helicopter, with the front bubble of a Huey helicopter, with the plastic windows, or more like a deep sea submarine, with a thick piece of glass bubble shaped, and where the tail rudder should have been, it was a black, egg-shaped pancake, and instead of landing gear it had upside-down rams horns that went from the top to the bottom and rested on the ends of the horns".
The description leaves people baffled. On Reddit, a user points out that the object described reminds of the Nellis UFO. I then find Powell's article and, with such a crazy shape, there's no way this is a coincidence, the two objects are the same. Between Powell's research and Grusch's witness, we can exclude that's a balloon or random debris, it must be secret technology. I can rule out that an alien decided to fly his machine in an area coincidentally famous for hosting secret and/or alien technology, so we're left with two possibilities: 1) that was human technology flown remotely or directly by humans, or 2) that was alien technology flown remotely or directly by humans.
 
Tardis UFO: The sources for this are Daniel Sheehan and Grusch, both reporting what one of Grusch's witnesses said. The Tardis is a fictional time machine and spacecraft, shaped like a phone box, which appears in the series Doctor Who.
The story involves a saucer partially embedded in the earth, with some fantastical properties. Sheehan said that "they tried to hook a bulldozer to it to pull it out. And it pulled out a shape like a pie slice, almost like it was part of the way it was constructed. When it came loose a couple of feet, they stopped immediately. They didn't want to destroy the integrity of the machine. They had a guy go into it. He got in there, and it was as big as a football stadium. It was freaking him out and started making him feel nauseous, he was so disoriented because it was so gigantic inside. It was the size of a football stadium, while the outside was only about ten metres in diameter. And space was not the only warped dimension around the craft. He staggered back out after being in there a couple of minutes, and outside it was four hours later. There was all kinds of time distortion and space distortion".
Million dollar question: you enter an alien spaceship as large as a stadium - and if you can tell how big it is, it means that you can see most of it, it's not like you're in complete darkness - and all you report is the spacetime distortion? Was that an empty stadium?

Landmark UFO: Ross Coulthart heard from "multiple sources" that there's a UFO that's so big that it couldn't be moved, "so big they built a building over it" (YouTube). It's not in the United States. Articles online quote him saying that the UFO is hidden beneath one of the world's major landmarks - hence the name I used for it - but I can't find the source for that exact quote, on YouTube he doesn't talk about landmarks. I thought this could be the same as the Tardis UFO. After all, how many giant unmovable UFOs can there be? But that was just a thought.

Triangle UFO: Two of Grusch's witnesses say that they saw a triangle-shaped craft, and one witness says that this particular UFO was successfully reverse-engineered. Triangle UFOs have a long history (Wikipedia), and "classified military aircraft may be responsible for many black triangle UFO reports" (Wikipedia). More than one possibility here: the triangle UFO has been reverse-engineered to produce non-triangular craft; the triangle UFO has been reverse-engineered to produce triangular craft; triangular craft are secret human technology and the UFO story is a cover-up or a misunderstanding.  


Grusch said that there are about fifty "gatekeepers" in the DoD who are keeping the program secret. In his report, he gives their names. These gatekeepers also control what AARO can know and say. For sure we know the names of the people who have been actively trying to prevent Congress from knowing more:

Lloyd Austin, Secretary of Defense.
Mike Turner (R), Chair of the House Intelligence Committee.
Mark Warner (D), Chair of the Senate Intelligence Committee.
Mike Rogers (R), Chair of the House Armed Services Committee.
Mitch McConnell (R), Senate Minority Leader.
Mike Johnson (R), Speaker of the House.

But these are not all necessarily in the know about the retrieval program. They might have been "warmly invited" to hinder the Congress, or they could have personal political reasons to do so.

Also:
Since 1946 the CIA has had 26 directors.
Since 1947 the DoD has had 28 secretaries.
Since 1980 the JSOC has had 17 commanders.
Did they all know about the retrieval program?