This course is designed to provide students with a sampling of mathematical delicacies, interesting and unusual thoughts that have been developed over tens of centuries. We shall follow the work of mathematicians, beginning with the ancient Greeks, who attempted to come to terms with the concept of infinity. We shall address mathematical questions about how large things are, how many, how fast, how often, as well as the amazing discovery that such questions do not always make sense. Paradoxes will be discussed, both in apparent forms and in irrefutable guises. We shall play mathematical games that will require us to learn something of probabilities and that, in turn, will require us to learn when to count and when not to count. We shall also discuss the personalities and motivations of great mathematicians through their biographies and autobiographies. The course aims to sharpen students' intellect by challenging them with problems in which the recognition of ideas is central. Students' imagination will be stimulated, and they will be encouraged to ask questions in areas about which we know little or nothing. Above all, students will marvel at the wonderfully surprising world of mathematical thought.