Director Mueller, 72, was named yesterday by the Justice Department to probe alleged Russian efforts to sway November's presidential election in favour of Donald Trump and to investigate whether there was any collusion between Trump's campaign team and Moscow.
President Trump said in a statement there was no collusion between his campaign and “any foreign entity”.
Director Mueller is known by some as “Bobby Three Sticks” because of his full name – Robert Mueller III – a moniker that belies the formal bearing and no-nonsense style of the former Marine Corps officer who was decorated during the Vietnam War.
Democrats and Republicans alike praised his appointment and hailed his integrity and reputation.
US Justice Department appoints prosecutor to probe ‘Russian efforts to sway election’
Director Mueller was named to the post by Deputy Attorney General Rod Rosenstein. His investigation will run in parallel to those being carried out by the FBI and Congress.
It would be difficult to fire Director Mueller, and past special counsel appointments have shown that the job comes with independence and autonomy.
Chicago federal prosecutor Patrick Fitzgerald was appointed during the George W. Bush administration in 2003 to a similar role to investigate the leak of the identity of Valerie Plame, an undercover CIA officer whose husband had criticised Bush administration policies.
Mr Fitzgerald indicted I. Lewis “Scooter” Libby, a top aide to Vice President Dick Cheney. President Bush granted Mr Libby clemency from a prison sentence before he left office.
Mr Trump said in a statement there was no collusion between his campaign and “any foreign entity”
Robert Luskin, a Washington lawyer who defended Bush political adviser Karl Rove during Mr Fitzgerald's investigation, praised the choice of Mr Mueller.
“I think it's good across the board,” Mr Luskin said. He described Director Mueller as “credible” and “independent” and said his appointment would be “good for the Department of Justice”.
Director Mueller, known for avoiding political controversy, took a stand in 2004 when he and then-deputy attorney general James Comey threatened to resign when the Bush White House sought to reauthorise a domestic wiretapping programme that the Justice Department had deemed unconstitutional.
In his new role as special counsel, Director Mueller will have wide latitude to take the investigation wherever he thinks it should go and can use the full range of Justice Department investigative tools, said Jack Sharman, an attorney who served as special counsel during a probe into the Whitewater real-estate investments of President Bill Clinton in 1995.
Director Mueller is best known for his no-nonsense style
Mr Mueller was appointed director of the Federal Bureau of Investigation by George W. Bush a week before the September 11, 2001, attacks on New York and Washington, DC.
He was credited with transforming the FBI after Congress and an independent government commission established that the agency and the CIA had failed to share information before the attacks that could have helped thwart them.
At the FBI, Director Mueller put more resources into counterterrorism investigations and improving its cooperation with other federal agencies.
Director Mueller was chief of the Justice Department's criminal division before becoming FBI director. Among his most famous cases was the fraud and racketeering indictment of individuals associated with a politically connected Luxembourg bank run out of London and led by bankers from Pakistan and Abu Dhabi.
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He also oversaw the investigation into the Pan Am airline bombing over Lockerbie and the drug case against former Panamanian dictator Manuel Noriega.
Director Mueller served 10 years as FBI chief under President Bush and his Democratic successor Barack Obama, who then signed legislation to extend Mr Mueller's FBI term for another two years.
He was succeeded as FBI director by James Comey, who was fired by President Trump last week.
In the wake of his firing, there have been media reports that President Trump had asked Director Comey in February to end an FBI investigation into the president's former national security adviser, Michael Flynn. The reports cited a memo said to have been written by Director Comey documenting President Trump's request.
There have been calls for months for a special counsel to oversee the investigation into any ties between President Trump's campaign team and Moscow.