Computer hacker Gary McKinnon's mother Janis Sharp said she was "overwhelmed" after a 10-year battle against extradition which had been "an emotional rollercoaster".
London mayor Boris Johnson said: "At last, justice and the well-being of Mr McKinnon have prevailed.
"I have long supported Gary's right to be tried here in the UK.
"To extradite a man diagnosed with Asperger syndrome to America for trial would have been extraordinarily cruel and inhumane. I applaud the Government's stance.
"If they had approved extradition, they'd have been saying that extradition on any grounds was OK. It's not."
Melanie Riley, of campaign group Friends Extradited, said: "We are relieved and delighted for Gary and his family and his lawyers.
"A brave and correct decision has finally been made by the Home Secretary."
Gary McKinnon's mother Janis Sharp thanked Home Secretary Theresa May for blocking her son's extradition, saying: "Thank you Theresa May from the bottom of my heart - I always knew you had the strength and courage to do the right thing."
Computer hacker Gary McKinnon won his 10-year fight against extradition today after Home Secretary Theresa May stepped in to halt proceedings.
Mrs May stopped his extradition on human rights ground after medical reports showed the 46-year-old was very likely to try to kill himself if extradited.
McKinnon was accused by US prosecutors of "the biggest military computer hack of all time", but he claims he was simply looking for evidence of UFOs. Mrs May said the sole issue she was considering was whether "extradition to the United States would breach his human rights".
There was no doubt McKinnon is "seriously ill" and the extradition warrant against him should be withdrawn, she said.
It is now for the director of public prosecutions Keir Starmer QC to decide whether he should face trial in the UK, Mrs May added.
Computer hacker Gary McKinnon will not be extradited to the United States, Home Secretary Theresa May said today.
Home Secretary Theresa May is due to make a statement to the Commons at 12.30pm, about whether or not computer hacker Gary McKinnon will be extradited to America.
Gary McKinnon's mother Janis Sharp says that her son has been suffering mental torture, while waiting to find out if he'll be extradited to the USA.
Clip here to watch a video clip of her talking to ITV news.
Gary McKinnon's local MP says that the Home Secretary will have "blood on her hands" if she approves his extradition to the USA.
David Burrowes, the MP for Enfield South, has also threatened to resign from his position as a ministerial aide to the Environment Secretary if the extradition goes ahead.
Click here to watch a clip of him talking to ITV Daybreak.
Gary McKinnon's mother Janis Sharp says that extraditing him to the USA would prove a "death sentence" as he is at a high risk of suicide.
Yesterday, she posted the following messages on Twitter.
@wsjaeboo Feeling very nervous.