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In a letter to party president Simon Hughes he said questions about his leadership were "getting in the way of further progress by the party".
Mr Hughes said the party owed Sir Menzies "a huge debt of gratitude".
Deputy leader Vincent Cable will take over as acting leader for now - a leadership timetable will be announced on Tuesday.
The official announcement was made by Mr Cable and Mr Hughes, who said Sir Menzies had taken the decision in the "interests of the party and of Liberal Democracy".