This very scholarly article surveys the appearance of thin sets in research on trigonometric, Rademacher, and Walsh series. It contains new results, new interpretations of old results, and a chart which gives complete answers to ten basic questions about thin sets. It also ends with several open problems and unanswered questions. One small drawback. Although the authors and journals in the six page bibliography have been translated into English, the titles of most of the Russian papers have not.