Abstract: Towards the end of the 19th century, H. Schubert developed a powerful formal calculus for enumerating curves in the plane and 3-space satisfying a range of geometric conditions (trivial example: how many lines contain each of two distinct points?). Recognizing that these techniques were not rigorous, the 15th Hilbert problem posed at the turn of the 20th century was to justify Schubert's methods. While some of Schubert's methods can now be completely understood in terms of cohomology and intersection theory, not all of Schubert's assertions are fully proven. Nevertheless, the search for rigor has led to the creation of major subfields of modern algebraic geometry which are currently active and thriving fields of research more than a century later. In a similar development, at the end of the 20th century, physicists developed new and completely unexpected methods for solving enumerative problems using string theory and quantum field theory. While these methods are far from rigorous, the results of "correctly done" physical calculations have always turned out to be true whenever they can also be answered by rigorous mathematical analysis. Once again, we have failed miserably so far in providing rigor to the physicists' mysterious methods, but as a result of the effort, new mathematics has been and is being created to solve previously unsolved problem in this area. Even better this time, enumerative methods in mathematics and physics continue to develop in parallel, each field challenging and inspiring the other to ever- more astounding discoveries. In this talk, I will convey a sense of the field through examples that do not require advanced mathematics to formulate.