The Daily Mail and Mail on Sunday publishers wanted to stop the case, arguing claims of getting "information by deception" were out of time.
But a judge has decided the case, involving Prince Harry and six other high-profile claimants, can proceed.
Associated Newspapers strongly denied the allegations as "preposterous".
In a High Court ruling on Friday, Mr Justice Nicklin said Associated Newspapers had "not been able to deliver a 'knockout blow' to the claims of any of these claimants".
As well as Prince Harry, the newspaper group faces multiple claims of "gross breaches of privacy" from Sir Elton John, David Furnish, Elizabeth Hurley, Sadie Frost, Sir Simon Hughes and Baroness Doreen Lawrence.
This includes allegations of bugging devices in cars, listening into phone calls and dishonestly obtaining medical and financial information.
"We are delighted with today's decision which allows our claims over serious criminal activity and gross breaches of privacy by the Mail titles to proceed to trial," said a joint statement from the claimants.
But a statement from Associated Newspapers said: "As we have always made unequivocally clear, the lurid claims made by Prince Harry and others of phone-hacking, landline-tapping, burglary and sticky-window microphones are simply preposterous and we look forward to establishing this in court in due course."
The newspaper group also welcomed as a "significant victory" the ruling that unpublished information given on "strict grounds of confidentiality" to theLeveson Inquiry into phone hackingcould not be used as evidence in this case.
Prince Harry, in this latest battle with the UK's tabloid press, made a surprise appearance at an earlier hearing of the the case against Associated Newspapers, at the High Court in London in March.
The Daily Mail and Mail on Sunday publishers had categorically denied the allegations and said the claims had "no real prospects of succeeding".
But their lawyers had also argued that in any event the allegations were outside the requirement to bring claims within six years.
Some of the allegations are from decades ago, but lawyers for Prince Harry and the claimants successfully argued that new evidence had come to light and they were unaware at the time of how information was being covertly acquired.
Mr Justice Nicklin's ruling accepted that the claimants might not have known about such "concealed" gathering of information about them.
"In my judgment, the claimants have a real prospect of demonstrating not only that the unlawful acts themselves were concealed, but also, in many instances, further devices were employed in the published articles to throw the subject 'off the scent'," said the judge's ruling.
"Several claimants complain that they believed that their confidences were being betrayed by people close to them."
The judge's decision was welcomed by actor Hugh Grant, the director of the Hacked Off group, which campaigns for press reforms.
"This ruling is a significant blow to the Daily Mail and great news for anyone who wants the truth about allegations of illegal press practices to come out," he said.
Privacy lawyer Philippa Dempster, of the law firm Freeths, said the ruling that the claims can go ahead despite the passage of time would "send a shockwave across the press industry".
The ruling could also mean another in-person court appearance from Prince Harry, who earlier this year stepped into the witness box to give evidence in a hacking claim against another newspaper publisher, Mirror Group Newspapers.
He became the first senior royal in modern times to make such a court appearance, facing questions over two days, with the outcome of that case still to be decided.
The Duke of Sussex can go ahead with claims against Associated Newspapers of unlawfully obtaining information, as a court ruling opens the way for a trial.