This week, she is asking for ANL’s defence to be thrown out as “utterly fanciful” which would deliver her a spectacular victory.
What is the case about?
At the centre of the bitter legal battle is a letter written by the Duchess in August 2018, sent to her father at his Mexico home.
The five-page, 1,250-word handwritten letter was sent through a trusted contact who knew her father, discussing their broken relationship and pleading with him to stop talking to the media.
Her lawyers say the contents of the letter were “intrinsically private, personal and sensitive”, touching on her love for him, concerns over his health, and her “sense of betrayal over his denial of working with the paparazzi”.
Just before Meghan’s wedding to Prince Harry, Mr Markle was accused of staging paparazzi photos for money when he was due to walk his daughter down the aisle.
In legal papers, the Duchess’ lawyers say the final line of the letter was: “If you love me, as you tell the press you do, please stop. Please allow us to live our lives in peace.”
How did the letter get out?
The existence of the letter was first revealed in an eight-page article in US magazine People, when a close friend said Meghan had written to her father asking him to stop “victimising” her.
Three days later, on February 9, 2019, Mail on Sunday and MailOnline articles were published, containing substantial extracts of the letter.
A further four articles were published by the media group, which Meghan say amounted to breaches of her privacy.
The 39-year-old claims publication of parts of the letter amount to misuse of private information, a breach of copyright, and intrusion into her private life.
She says the message to her father was never intended for public consumption and contained intensely personal and private information.
She revealed the pain she was suffering as a result of the breakdown of her relationship with her father, poured out her heart to him, and issued a healtfelt plea for him to stop cooperating with negative press stories.
She launched the High Court claim in October 2019, setting up a legal battle which has made headlines around the world.
The Mail on Sunday’s defence
The newspaper publisher argues Meghan knew her letter would be read by people other than her father - including possibly journalists - and they say it was written with that firmly in mind.
It argues the Duchess did not have an expectation of privacy, and there is a strong public interest in the activities of the Royal Family.
The newspaper group also contends that its stories were needed to set the record straight after the People magazine article, by putting across Thomas Markle’s side of the story.
Adding into the case last year was a biography of Meghan and Harry, Finding Freedom, which touched on the letter controversy.
Meghan says she allowed a friend to pass on personal information to the authors of the book – to set out the “true” story – while the Mail suggests this shows she was willing to reveal details of her private life.
What is happening at court this week?
The Duchess is seeking summary judgment in her case, arguing ANL’s defence is hopeless and does not stand a chance of succeeding.
Her lawyers have called the defence “prolix, diffuse, and lacking in clarity”, asking Mr Justice Warby to end the case and hand her victory.
ANL is fighting the legal move, arguing this is a case that can only be resolved with a full trial.
What happens next?
If the judge refuses to stop the case, a trial is due to take place in autumn this year.
The case was due to be heard this month, but was delayed after a request by Meghan which ANL did not oppose.
Meghan has brought this legal claim to protect her privacy. But if the case goes to trial she is likely to find her personal life under even more global scrutiny.
Thomas Markle is expected to be among the witnesses for ANL, and Meghan herself would likely have to give evidence and face questions about her personal dealings.
Mr Justice Warby will hear the summary judgment argument over two days and is likely to deliver his judgment at a later date.