The 40-year-old returns to England after a brief 123-day spell in charge of Rangers in the Scottish Premiership, which ended in October, as he looks to rebuild his coaching career.
Martin takes over a Leicester side now competing in League One following back-to-back relegations.
He replaces Gary Rowett, who was unable to keep the Foxes in the Championship last season after taking over a squad already impacted by a six-point deduction for previous financial breaches.
"This is a club with great history, strong support and high expectations, and I'm looking forward to getting to know the club, the city and the supporters," said Martin per BBC.
"My immediate focus is on the team: building strong relationships, setting clear standards and creating performances that Leicester City supporters can connect with and be proud of."
Martin becomes the seventh permanent manager to take charge of Leicester since Brendan Rodgers’ departure in April 2023 following their FA Cup triumph.