Madeleine McCann's mother Kate said today she hoped the Leveson report would "mark the start of a new era" for the press, in which it treats those in the news "with care and consideration".
Mrs McCann went on: "Needless to say, more time will be needed to read, digest and understand all the implications of the report but initial impressions are positive.
"I hope the Prime Minister, and all the party leaders, will embrace the report and act swiftly to ensure activation of Lord [Justice] Leveson's recommendations within an acceptable and clearly defined time-scale."
The Hacked Off campaign, which represents many victims of newspaper intrusion, said Lord Justice Leveson had given the Prime Minister a "workable, proportionate, reasonable solution" to the problems of press abuse.
But the group added that David Cameron "had not done his job" by failing to accept the full recommendations of the report and was wrong to trust Editors to "behave themselves".
The media lawyer Mark Lewis, who represents the family of murdered schoolgirl Milly Dowler, said he and the Dowlers are "cautiously optimistic" about Lord Justice Leveson's report.
He said he hoped that the MPs who signed an open letter opposing the statutory underpinning of the press regulator would look again at the proposal.
Dominic Crossley, who represented the families of Milly Dowler and Madeleine McCann during the inquiry, spoke of the victims and said:
They want a better and more responsible press because they recognise its value.
They are just a small selection of the hundreds of other people who have been the victims of spears, bullies and attention.
David Sherborne, who represents the core participant victims of phone hacking, said his clients welcome the findings and recommendations of the Leveson report.
He said the report included a "clear recognition of widespread failings in the culture, ethics and standard of the press and the devastating consequences this has had for the victims".
He said that the report calls for a regulatory system with "real teeth" and called on the current parliament to implement its recommendations.
The former Formula One chief Max Mosley has said that the recommendations set out in the Leveson report "would make the situation much better than it is now".
He added that it would be "astonishing" if MPs chose not to implement the recommendations since "no responsible politician could allow the current situation to continue".
Actor Hugh Grant has tweeted that he, and other phone hacking victims, are about to see his advance copy of the Leveson report.
With victims about to get our advance copy of Leveson's report. Read my blog on press abuse myths while I'm locked in http://t.co/YM0c5vur
The parents of the murdered schoolgirl Milly Dowler, Bob and Sally, have arrived at the QE2 centre in Westminster ahead of Lord Justice Leveson's statement this afternoon.
It was the revelation that their daughter's voicemail messages had been hacked that led to the public outcry over the practice, and ultimately the Leveson inquiry.
The media lawyer Mark Lewis arrived at the same time.