The decorated Marine and Vietnam War veteran also documented Russia's interference in the 2016 US election and its contacts with Donald Trump's campaign but opted not to bring criminal charges against a sitting president.
"With deep sadness, we are sharing the news that Bob passed away" on Friday night, his family said in a statement Saturday.
"His family asks that their privacy be respected.".
The New York Times last year reported that Mueller had Parkinson's disease.
Trump on Saturday celebrated Mueller's death.
"Good, I'm glad he's dead," Trump wrote on the Truth Social site.
"He can no longer hurt innocent people!"
Former Democratic President Barack Obama called Mueller "one of the finest directors in the history of the FBI" who saved "countless lives" after transforming the bureau.
"But it was his relentless commitment to the rule of law and his unwavering belief in our bedrock values that made him one of the most respected public servants of our time," Obama said on X.
Mueller retired after 12 years as Federal Bureau of Investigation director in 2013 but was summoned back to public service by a senior Justice Department official four years later as a special counsel to take over an inquiry into Russia's election meddling after Trump fired then-FBI chief James Comey.
Mueller conducted a 22-month investigation that produced indictments against 34 people, including several Trump associates as well as Russian intelligence officers and three Russian companies, and a series of guilty pleas and convictions.
All told, Mueller brought criminal charges against six of the president's associates, including his campaign chairman and first national security adviser.
Mueller ultimately stopped short of a criminal indictment of the Republican president, bitterly disappointing many Democrats.
Mueller was a Princeton graduate who walked away from a lucrative midcareer job to stay in public service, and his old-school, buttoned-down style made him an anachronism during a social media-saturated era.
During his career as a prosecutor and FBI chief, Mueller displayed a patrician manner and sometimes wooden personality - just about the opposite of the bombastic Trump.
He was known by some as "Bobby Three Sticks" because of his full name - Robert Mueller III - a moniker that belied his formal bearing and sober approach to law enforcement.
His Russia inquiry, detailed in a 448-page 2019 report, laid bare what Mueller and US intelligence agencies have described as a Russian campaign of hacking and propaganda to sow discord in the US, denigrate 2016 Democratic presidential candidate Hillary Clinton and boost Trump, the Kremlin's preferred candidate. Russia denied election interference.
"First, our investigation found that the Russian government interfered in our election in sweeping and systematic fashion," Mueller said during 2019 congressional testimony.
"Second, the investigation ... focused on whether the evidence was sufficient to charge any member of the (Trump) campaign with taking part in a criminal conspiracy. It was not," Mueller added.
"The president was not exculpated for the acts that he allegedly committed."
Mueller, a longtime Republican, faced unremitting attacks by Trump and his allies on his integrity as they tried to discredit the investigation and the special counsel himself. Trump used social media, speeches and comments to news media to assail Mueller, accusing him of running a politically-motivated "rigged witch hunt".
"It's all a big hoax," Trump said in 2019.
"Absolutely, it was not a hoax," Mueller told the congressional hearing, noting the numerous charges arising from the probe.
Appointed by Republican President George W Bush to head the FBI, Mueller took over as its director one week before the September 11 attacks on the US by al-Qaeda militants using hijacked planes that killed about 3000 people.
Obama later extended Mueller's appointment. By the time Mueller left the position, his tenure was exceeded only by J Edgar Hoover's 48-year stint.
with AP
