Volkswagen has agreed to recall and modify the majority of the 11 million diesel cars known to have the diesel emissions-cheating software it admitted to four weeks ago.

This week, Germany's Federal Motor Transport Authority (KBA), approved a timetable and proposed fixes submitted by VW, allowing the company to move forward with a recall in that country.

Volkswagen says it will recall a total of 8.5 million TDI models in Europe that are equipped with the EA189 four-cylinder engine, including a confirmed 2.4 million just in Germany.

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Other European countries will clarify in turn exactly which EA189-equipped models within their jurisdictions are affected.

Volkswagen will contact European customers with affected cars directly, and is setting up websites so customers can check to see if their cars are being recalled.

It plans to begin implementing fixes in January, at no cost to customers.

2015 Volkswagen Golf TDI SE

2015 Volkswagen Golf TDI SE

The company is not discussing exactly how it will address the emissions issue, either in Europe or in North America, but says fixes can involve "software as well as hardware measures," depending on the model.

This week's agreement to mount a massive recall across Europe does not include approximately 482,000 similar vehicles in the United States that have been identified as having the same emissions "defeat device" software.

ALSO SEE: Why Did Volkswagen Cheat On Diesel Emissions In Its TDI Cars?

All cars set to be recalled so far in Europe use the EA189 engine, which is also found in the majority of affected cars in the U.S.

That includes 325,000 Jetta, Jetta SportWagen, Golf, Beetle, and Audi A3 TDI models sold during model years 2009 through 2015.

2014 Volkswagen Beetle TDI

2014 Volkswagen Beetle TDI

None of these models have the Selective Catalytic Reduction (SCR) urea-injection system that all other manufacturers use to comply with U.S. emissions standards in force since 2008.

It's possible the affected VW models will need to have SCR systems installed or undergo software changes to comply with Environmental Protection Agency rules on emissions of nitrogen oxides (NOx).

VW is also reportedly considering buybacks or cash payments for owners in the U.S.

MORE: VW Diesel Emissions Recall: What You Need To Know In 10 Questions

Some 90,000 Passat TDI sedans in the U.S. will have to be recalled as well; these models use the same EA189 engine, but with an SCR system and Lean NOx Trap, which should mean less-extensive modifications.

In the meantime, Volkswagen continues to stress that all affected cars worldwide remain safe to drive, and that owners should continue to do so.

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