The appeal in Paris concerns photos of Kate and Prince William on holiday in the south of France, which were published in French Closer and La Provence magazines in September 2012.
They were snapped with a zoom lens as they relaxed outside at a chateau belonging to the Queen’s nephew Viscount Linley.
Closer Magazine was ordered to pay €100,000 (£92,000) in damages to Kate and William over the breach of privacy last year.
However, lawyers for the magazine claim the compensation amount was “exaggerated” due to their royal status.
They will also argue that younger members of the royal family now invite high-levels of media scrutiny due to the fact they increasingly act like celebrities.
Screen shots from a 90-second video featuring Meghan, which was filmed in 2013 for Men’s Health Magazine, will be shown to the court as part of their evidence.
During the video called The Ultimate Guy’s Girl, the now-Duchess of Sussex strips down from a dress and jacket before firing up the barbecue and tucking into a burger.
At the time of filming, her acting career was blossoming and she was best known for her role as lawyer Rachel Zane in Suits.
Meghan has now given up her acting career in order to focus on royal duties.
Lawyers for Closer are asking for damages to be reduced in line with routine privacy cases, which usually amount to around £100.
The publication’s editor, Laurence Pieau, 51, and owner Ernesto Mauri, 71, also had to pay fines of €45,000 (£41,000) – the maximum possible – which they dispute.
At the time of the original case, Prince William told the court in a statement the case was “all the more painful” because of the way the paparazzi hounded his mother before her death in the French capital 20 years ago.
The Duke and Duchess of Cambridge were originally seeking $1.5m (£1.3m) in damages.
The appeal is due to start in the Versailles Appeal Court.