Intersection theory is a big subject that has played an important role in algebraic geometry, and any attempt at a comprehensive introduction in 90 minutes would surely fail. With this in mind, I have decided instead to attempt to convey some of the beauty and flavor of intersection theory by way of discussing a few concrete classical examples of intersection theory on surfaces. The material in the talk is covered in almost any text in algebraic geometry. There are many approaches to finding 27 lines on a cubic surface. If one wishes to see a more rigorous version of the presentation given in this talk, then please see Hartshorne's treatment in Chapter V.4 of the book cited in the refereces. Chapter V of Hartshorne's book is a very nice introduction to intersection theory on surfaces. For a more general orientation in the subject with good historical context, one might wish to read Fulton's "Introduction to intersection theory in algebraic geometry." Fulton's other book cited in the references is the standard in the subject, and the other texts listed offer additional points of view.