As mathematics becomes ever more complex, more structured, and more specialized, it is easy to lose sight of the fact that the research frontiers still contain many unsolved problems that are of immediate intuitive appeal and can be understood very easily, at least in the sense of understanding what a problem asks. Parts of plane geometry and number theory are especially good sources of such intuitively appealing problems, and those are the areas emphasized in this book. For each of the presented problems, we look at its background, try to explain why some mathematicians have found the problem of special interest, and tell about techniques that have been used to obtain partial results. Some of the exercises will help out readers establish partial results for themselves, and references are provided for those who want to learn more about the problems.