Barr's letter was summary of what he called the "principal conclusions" of Mueller's high-anticipated report on his investigation into possible collusion between the Trump campaign and Moscow during the 2016 election and whether the president obstructed justice, among other things.
The four-page document said special counsel Mr Mueller did not find that Mr Trump's campaign "conspired or co-ordinated" with the Russian government to influence the 2016 presidential election, knocking down arguments from Democrats who have claimed there was evidence of such collusion.
"The fact that Special Counsel Mueller's report does not exonerate the president on a charge as serious as obstruction of justice demonstrates how urgent it is that the full report and underlying documentation be made public without any further delay", House Speaker Nancy Pelosi and Senate Democratic Leader Chuck Schumer said in a joint statement.
"After 22 months of a special counsel and 2 years of congressional investigations, it's over", tweeted North Carolina Rep. Mark Meadows, a close ally of Trump.
Trump pledged new investigations but did not specify who would conduct them or who should be targeted.
During his investigation, Mr Mueller looked at whether a number of Mr Trump's actions might have obstructed the course of justice.
Trump, who has spent this weekend at his private club in the state of Florida, touted Barr's summary as a "complete exoneration", while again lashing out at Mueller's investigation that has shadowed his presidency for the past two years, which he has repeatedly called a "hoax" or a "witch hunt".
Mr Trump, asked on Monday if he would be OK with the release of the full report, responded: "Up to the attorney general, but it wouldn't bother me at all".
Trump has spent months railing against former Justice Department officials, including former FBI Director James Comey, accusing them of an illegal witch hunt for the goal of delegitimizing his presidency.
"It's one of those hard realities that sometimes things don't turn out the way you wanted them to", said Democratic political consultant Colin Strother, who is based in Austin, Texas.
Barr signaled in his letter that more information of the Mueller report may be released in the future. They include several of his former aides such as former campaign chairman Paul Manafort, former national security adviser Michael Flynn and former personal lawyer Michael Cohen. "Democratic leaders were already backing away from it", said Elaine Kamarck, a senior fellow at the liberal-leaning Brookings Institution in Washington. The special counsel also detailed extensive contacts between Trump associates and Russian Federation.
Mueller wrapped up his investigation after almost two years on Friday and submitted his findings to Attorney-General William Barr, who on Sunday released a four-page summary saying there was no evidence of criminal collusion between Trump's campaign and Russian Federation in the 2016 election.
Democrats won control of the U.S. House of Representatives in congressional elections last November and have used their power since January to launch blockbuster hearings in the powerful House Judiciary and House Oversight committees. Despite lengthy negotiations, Mueller never obtained an in-person interview with the president.
All they needed was a stinging condemnation from Mueller of Trump's behavior to give their probes liftoff.
The Kremlin said on Monday that President Vladimir Putin was ready to improve ties with the United States following the release of Barr's summary and called on Washington to formally recognise there was no collusion.
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