Mueller, a Vietnam war veteran who served as director of the FBI from 2001 to 2013, is described as enjoying seamless respect from Democrats and Republicans alike as he undertakes a high-stakes mission which could determine the fate of Donald Trump's presidency.
"Mueller is a great selection. Impeccable credentials. Should be widely accepted," tweeted Jason Chaffetz, the Republican chairman of the House Government Oversight Committee.
The New York-born Mueller boasts "determination and independence of mind," added Democratic Senator Richard Blumenthal, one of Trump's loudest critics.
"I think that Bob Mueller is the kind of prosecutor and investigator we need here," Blumenthal added.
Trump's opponents are suggesting the president may have committed obstruction of justice -- an impeachable offense -- by allegedly asking FBI director James Comey in February to drop an investigation related to the Russia probe before sacking him last week.
Mueller, 72, is known for a no-nonsense management style developed during a tenure that began with him firmly in the hot seat: he took up the FBI post just a week before the September 11, 2001 terrorist attacks in New York and Washington.
Much of his work then centered on saving and reforming the Federal Bureau of Investigation amid calls for its dismantling after the agency missed clues that pointed to a massive attack against America being in the making.
Under Mueller, the FBI was rebooted into an agency with the new task of countering terror threats, and became a key part of the American national security apparatus.
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Showdown with a president
One of Mueller's many accomplishments was the foiling of an Al-Qaeda attack on a commercial airliner bound for Detroit on Christmas Day 2009.
Mueller proved his mettle famously in a dramatic 2004 showdown in which he threatened to resign rather than go along with reauthorization of a post-9/11 domestic spying program under then president George W. Bush.
The Justice Department, which oversees the FBI, had determined that it was illegal.
Mueller and his eventual successor -- Comey, then deputy attorney general -- backed down when the Bush administration agreed to make changes to the program, which was eventually disclosed to the public in 2005 by the New York Times.
In the end, Mueller was in the top FBI post for the second longest period after J. Edgar Hoover, who held it for 48 years until his death.
