Many mathematicians have viewed problem solving as the most important aspect of our discipline. The problems sections in today’s mathematics periodicals are a continuation of a longstanding practice in which mathematicians have challenged other mathematicians. Fibonacci, the Italian algebraists and Newton are classic examples. More recently, Erdos challenged audiences, offering monetary prizes. All MAA journals, and many others, have problems sections in which specific problems are posed and the “best” solutions published. We’ll review the history and provide many examples and observations.