This text was written as an antidote to topology courses such as Spivak describes. It is meant to provide the student with an experience in geometric topology. Traditionally, the only topology an undergraduate might see is point-set topology at a fairly abstract level. The next course the average student would take would be a graduate course in algebraic topology, and such courses are commonly very homological in nature, providing quick access to current research, but not developing any intuition or geometric sense. I have tried in this text to provide the undergraduate with a pragmatic introduction to the field, including a sampling from point-set, geometric, and algebraic topology, and trying not to include anything that the student cannot immediately experience. The exercises are to be considered as an integral part of the text and, ideally, should be addressed when they are met, rather than at the end of a block of material. Many of them are quite easy and are intended to give the student practice working with the definitions and digesting the current topic before proceeding. The appendix provides a brief survey of the group theory needed. The choice of topics addressed is my own, and fairly random at that, and was dictated by the fact that the text is used in a semester course. I often augment the text with articles from the general scientific and undergraduate math journals, which is a gentle way of getting the student to read and discuss mathematics.