(In September 13 story, corrects name of European Union Aviation Safety Agency from European Aviation and Space Agency in paragraph 15)

An unpainted Boeing 737 MAX aircraft is seen parked in an aerial photo at Renton Municipal Airport near the Boeing Renton facility in Renton, Washington, U.S. July 1, 2019. Picture taken July 1, 2019. REUTERS/Lindsey Wasson

CHICAGO (Reuters) - Boeing Co's BA.N best-selling jet, the 737 MAX, was grounded globally in March, days after the crash of an Ethiopian Airlines flight that followed a similar Lion Air disaster in Indonesia in October. A total of 346 people died in the two crashes.

Boeing has spent months working on updating critical flight control software at the center of both crashes, with the hope of winning Federal Aviation Administration approval for the planes to fly again in the United States between October and December.

That timeframe may differ in other countries.

Following are some of the key steps that need to take place before the 737 MAX returns to the skies.

SOFTWARE UPDATES AND FLIGHT TESTS

Boeing is still finalizing updates to flight control software, which will be followed by a certification test flight. CEO Dennis Muilenburg has said he was targeting the September timeframe for the flight but on Wednesday declined to give a specific date. Federal officials say it may not occur until October and the FAA may not give the green light to resume flights until November.

Boeing and FAA pilots have been testing the updated software for months and the FAA is inviting regular 737 MAX line pilots to run tests as well.

Those tests must also be completed before the aircraft is approved for return to service, the FAA has said.

PILOT TRAINING

The FAA’s Flight Standardization Board that determines U.S. pilot-training requirements must issue new recommendations for what kind of training is needed before U.S. airlines can fly passengers on the 737 MAX again.

The final report is expected in September and will have a 30-day period for comments from airlines and pilots.

A draft report in April recommended short computer-based training and classroom instruction about the new software. Simulator training, which some overseas regulators are weighing, would take longer and is more costly.

CERTIFICATION

Once the software updates, test flights and pilot training recommendations are finalized, the FAA must approve the jets for flight and has said it will need about 30 days from the time the certification flight is completed before it decides whether to allow flights to resume. The agency has repeatedly said it will not certify the plane until it is safe to do so.

The European Union Aviation Safety Agency said on Tuesday will conduct its own test flights separate from, but in full coordination with, the FAA. The test flights are not scheduled yet and depend on Boeing’s progress.

Muilenburg has said it is possible not all regulators around the world will concurrently approve the MAX to fly again.

Some officials do not expect the 737 MAX to actually resume flights until early 2020.

COMMERCIAL FLIGHTS

Without the 737 MAX, airlines that were flying the jets before its grounding have had to cancel flights, cut routes and use aging jetliners. Once the MAX is approved to fly again, airlines will have to install the new software, run a series of maintenance checks on the idled jets and implement the recommended pilot training.

U.S. airlines are hoping to fly the jets again later this year or early next year, pending regulatory approval.

Other airlines may have to await approval from their own regulators, some of whom have announced additional checks. China, a key importer which was the first country to ground the plane, has not said what if any additional checks it will carry out.

PROBES

Independently from the 737 MAX regulatory process, Boeing is facing numerous probes into the development of the aircraft by regulators, U.S. lawmakers and the Department of Justice. It also faces more than 100 lawsuits by victims’ families alleging it designed a flawed airplane. Some of them have demanded a full certification review of the 737 MAX.