Launched after the phone hacking scandal at the News of the World, the judge's eight-month inquiry heard testimony from various press intrusion victims, including Hugh Grant, Sienna Miller, Charlotte Church and the family of murdered schoolgirl Milly Dowler.
The resulting report was submitted to David Cameron yesterday, and there have already been reports that the prime minister is at loggerheads with his deputy, Nick Clegg, over the findings.
The report is released on the same day that former News of the World editor Andy Coulson and ex-News International chief executive Rebekah Brooks will appear in court to face charges related to alleged corrupt payments made to public officials.
Lord Justice Leveson is widely expected to recommend some form of statutory press regulation overseen by an independent body, similarly to the way Ofcom oversees the broadcast and telecoms industries.
The press is currently self-regulated by the Press Complaints Commission (PCC), but it has been widely criticised.
But a group of MPs and peers have warned that moving away from self-regulation could jeopardise free speech in the press, and disadvantage the print media over even less regulated internet news sources.
The office of House of Commons Speaker John Bercow has said that it will allow a request for the deputy prime minister to oppose Cameron at the despatch box after the prime minister's statement at 3pm - should they be unable to reach agreement.
A Coalition committee will meet this morning to try and thrash out a unified way forward.
Nick Clegg has said that the country needs a "raucous press" that holds people in power to account, but the newspapers must also be held to account for their worst excesses, as seen in the hacking scandal.
"That's the balance we need to strike," he said this morning.
It is thought that all parties want a new independent regulator with real teeth to replace the PCC, but the disagreement is over whether a new law is required to force all newspapers to sign up.
Yesterday, a ComRes poll for ITV showed that 51% of the public felt that the government should "introduce statutory regulation of the media", whereas 20% disagreed.
A separate ComRes survey for BBC Radio 5 Live indicated that 47% wanted "a regulatory body with rules agreed and enforced by the courts", whereas just 12% wanted a self-regulated system.
Lord Justice Leveson's report will be published at 1.30pm, and the judge will make a statement of his findings and recommendation at the Queen Elizabeth II centre in Westminster.
The Hacked Off campaign group - which has Hugh Grant on its board and has called for statutory regulation of the press - will hold a meeting at 4pm today, with some hacking victims due to attend.
Separately, Coulson and Brooks are among five people due to appear at Westminster Magistrates' Court today, alongside various other News International employees.
Former Downing Street Communications chief Coulson and ex-News of the World royal correspondent Clive Goodman are facing charges of conspiracy to pay for information and two counts of conspiring to commit misconduct in a public office.
Brooks is charged with conspiring to commit misconduct in a public office, alongside former Sun chief reporter John Kay and Ministry of Defence employee Bettina Jordan-Barber.