By Michael Pearson CNN

(CNN) -- It's been four days since EgyptAir Flight 804 disappeared off radar over the Mediterranean Sea with 66 people aboard, and questions about what happened to the flight abound.

Authorities have recovered some personal belongings, life vests, aircraft chairs and even body parts, but the bulk of the wreckage has so far eluded the teams searching for it.

And as searchers continue to look for the plane's fuselage and vital flight data and cockpit voice recorders, developments pour in.

Here's what you need to get up to speed:

French families voice concerns

A representative of an organization devoted to families of French passengers said the relatives aren't sure they can trust the Egyptian investigation and want more involvement from France.

"We need the French government in the diplomatic game to put diplomatic pressure and in the search and the identification of the bodies," said Stephane Gicquel, secretary general of the French National Federation of the Victims of Catastrophes.

"We want that the gendarmes can go and work with the Egyptian authorities. So I think there will be a way to track the investigation put in place by the French government with an appointed official to defend the interests of the families in the weeks and years to come," she said.

Gicquel said the families are distrustful of Egypt, in part due to how the country handled the investigation of the 2004 crash of a jet carrying mostly French tourists taking off from the resort town of Sharm el-Sheikh.

Egyptian investigators "cooperated very badly with the French authorities" at that time, she said.

For their part, Egyptian government and airline officials have promised to be transparent in the investigation into what happened to the plane.

Smoke detected

On Saturday, French aviation officials revealed that an automated system aboard the plane had sent messages about smoke in the front of the aircraft just before it crashed into the sea early Thursday en route from Paris to Cairo.

France's revelation confirmed flight data that CNN obtained from an Egyptian source. That flight data indicated smoke alerts occurred near the cockpit minutes before the crash.

The data came through the Aircraft Communications Addressing and Reporting System, or ACARS, which sends messages between planes and ground facilities. A screen grab of data has time stamps that match the approximate time the aircraft went missing.

There were indications of problems with a heated window in the cockpit, a sliding window in the cockpit, smoke in the lavatory, smoke in the avionics compartment below the cockpit, a fixed window, the autopilot and the flight control system.

The alerts don't necessarily mean a fire occurred on the Airbus A320 or that the crew even knew about the alerts, which are automatically transmitted, aviation experts cautioned.

There have been electrical problems with window anti-ice heaters in A320s. In 2003, the Federal Aviation Administration required windshields replaced in all A320s in the United States. It's not known whether Egypt followed the FAA directive.

Submarines deployed

On Sunday, Egypt tasked a submarine belonging to its Petroleum Ministry to look for the plane's flight data and cockpit voice recorders, the so-called black boxes. France sent a remotely piloted submarine of its own to help with the search.

The Egyptian submarine can reach depths of 3,000 meters (about 9,800 feet).

The French vehicle arrived Monday afternoon aboard the patrol ship Jacoubet, according to the French navy. The sub can dive up to 1,000 meters (3,280 feet) and is equipped with an acoustic probe made to detect signals from flight recorders.

The Egyptian military said Friday that it had spotted debris about 290 kilometers (180 miles) north of Alexandria, Egypt.

The search could take days or weeks, experts warn.

Debris and body parts will be taken to a military base and then to Cairo International Airport. EgyptAir has asked families to provide as much information as possible to identify body parts.

Recovered parts of the wreckage will be handed over to the state security prosecutor handling the criminal part of the investigation.

The theories

With little physical evidence, the possibilities about the plane remain wide open, from a bomb to a mechanical malfunction to pilot error or even an intentional crash.

No group has claimed responsibility for taking down the jet, and as of now, investigators have found nothing implicating the flight crew or security officials aboard the plane, an Egyptian official said.

Checks of the passenger manifest have so far resulted in no hits on terror watch lists, officials with knowledge of the investigation said.

The jet had routine maintenance checks in Cairo before it left for Paris, the airline said. Earlier Wednesday, the jet was also in Eritrea and Tunisia, data from flight-tracking websites show.

The Egyptian Civil Aviation Ministry has formed an investigative committee for the crash led by Ayman al-Moqadem, the investigator also in charge of the inquiry into the October crash of a Russian Metrojet airliner over the Sinai. That disaster, which killed all 224 aboard, is widely believed to be the work of terrorists.

A preliminary report of the investigation will be available within a month, Moqadem told state media Sunday.

On Monday, Egyptian state TV reported that Egypt's prosecutor general, Nabil El Sadeq, has asked Greek officials for transcripts of all calls between the EgyptAir crew and Greek air controllers.

Egypt has also asked France for information, including records, audio recordings and video concerning the plane's departure from Charles de Gaulle Airport in Paris until it left French airspace bound for Cairo, state media reported.

Hours after the flight disappeared, Egyptian Civil Aviation Minister Sherif Fathi told reporters Thursday that an act of terror was more likely to blame than a mechanical failure.

But he was more cautious Sunday, urging the media not to jump to conclusions or speculate about what caused the crash.

"In cases like this we need to wait until we base our judgment on facts," Fathi said in an interview with CNN's Becky Anderson. "We cannot at this stage come up with any conclusion."

French Foreign Minister Jean-Marc Ayrault told passengers' families Saturday that "no theory" had been ruled out.

CNN's Ralph Ellis, Ben Wedeman, Tiffany Ap, Catherine E. Shoichet, Joe Sterling, Hamdi Alkhshali, Madison Park, Laura Goehler, Ian Lee and journalist Sarah Sirgany contributed to this report.

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