The family of Michael Brown, the black teenager shot dead by a policeman, say they are "crushed" by the grand jury's decision not to charge Darren Wilson.Mr Brown's father told NBC news his son's character had been "crucified". Separately, Mr Brown's mother said Mr Wilson had been "disrespectful".
Mr Wilson, who shot dead Mr Brown in Ferguson, Missouri, in August, has told US media he has a "clean conscience".
There were protests over the ruling in 13 US cities on Tuesday night.
In Ferguson, there was some unrest on as protesters scuffled with police, and a police car outside the town hall was set alight.
However, police said security was "much better" than Monday, when there was widespread rioting and looting, and more than a dozen buildings were set alight.
About 2,000 National Guard troops were deployed in the city on Tuesday night. Forty-four people were arrested.
Protests were reported in 12 other cities: Philadelphia, Seattle, Albuquerque, New York, Cleveland, Los Angeles, Oakland, Minneapolis, Atlanta, Portland, Chicago and Boston.
Those demonstrations were mostly peaceful, but rioting broke out in Oakland, California, where protesters started a street fire.
In Los Angeles, 130 protesters were arrested for disorderly conduct, the LA Times reported.
'Insult after injury' Michael Brown was shot by a white police officer, Darren Wilson, on 9 August, sparking protests.
Speaking out after the verdict on Monday, Mr Wilson told US broadcaster ABC that he had a "clean conscience".
He said that before the shooting, Mr Brown had pushed him back into his car, hit him and grabbed at his drawn gun, and said that he felt "like a five-year-old holding on to [US wrestler] Hulk Hogan".
In a separate interview with CBS, she added: "I don't believe a word of it. I know my son far too well... Our son doesn't have a history of violence."
Meanwhile, Mr Brown's father, Michael Brown Sr, said that he felt "hurt" and "crushed" following the verdict.
He said that "terrible" things had been said about his son, and prosecutors had "crucified his character".
He also urged protesters to avoid violence, saying: "My son was a good guy, a quiet guy. So in his name, I want to keep it on a positive note."
Mr Brown's supporters said he was attempting to surrender to Mr Wilson when he was shot.
Many in Ferguson's predominantly African-American community had called for Mr Wilson to be charged with murder, but the grand jury - of nine white and three black members - decided not to charge him.
The decision means Mr Wilson will not face state criminal charges over the shooting. However, the US Justice Department has also launched a federal investigation into whether Mr Wilson violated Mr Brown's civil rights.
Speaking from Chicago on Tuesday, President Barack Obama said there was "no excuse" for destructive behaviour and criminal acts of rioting.
However, he added that "many communities of colour" had a sense that laws were not being enforced "uniformly or fairly", and said he had ordered Attorney General Eric Holder to look at what steps could be taken to build trust.