The announcements were welcomed by Kenny Dalglish, who was Liverpool's manager during the 1989 FA Cup semi-final at Hillsborough.
Two fantastic results today for the Hillsborough families. Your support has been unbelievable.From @kennethdalglish on Twitter:
Trevor Hicks, Margaret Aspinall and Jenni Hicks have spoken of their delight at the decision to quash the inquest verdicts for the Hillsborough victims.
They said they now hope for an unlawfully killed verdict.
Speaking outside London's Royal Courts of Justice, Mr Hicks said: "Justice is on its way.
"Everything we've said has been proven to be correct."
Trevor Hicks, who lost both daughters in the Hillsborough disaster, said the victims' families were "delighted" with today's ruling.
Trevor Hicks, who lost both daughters in the Hillsborough disaster, said the victims' families were "delighted" with today's ruling and added that "justice is on its way".
Outside the High Court, Mr Hicks added that today was a "huge step for the families".
The High Court has quashed the original accidental death inquest verdicts returned after 96 Liverpool football fans died in the crush at Hillsborough 23 years ago.
The Lord Chief Justice Lord Judge and two other judges in London ordered fresh inquests following an application by the Attorney General Dominic Grieve.
The Liverpool Echo are reporting that it took the High Court judges just three minutes to consider their Hillsborough inquests verdict.
Attorney General Dominic Grieve told the court: "This is an exceptional application but new inquests are needed in the interest of justice".