This elementary introduction to stable maps and quantum cohomology presents the problem of counting rational plane curves; the viewpoint is mostly that of enumerative geometry. Emphasis is given throughout the exposition to examples, heuristic discussions, and simple applications of the basic tools to best convey the intuition behind the subject. The book demystifies these new quantum techniques by showing how they fit into classical algebraic geometry. The book is ideal for self-study, as a text for a mini-course in quantum cohomology, or as a special topics text in a standard course in intersection theory. The book will prove equally useful to graduate students in the classroom setting as to researchers, geometers, and physicists working in the field.