Points and LinesAbstract: Suppose you take some points in the plane, and that they are not all on one line. Then there must be a line passing through precisely two of the points. This is not an obvious fact, and in fact it was an open problem for 50 years before being solved by Gallai in 1943. I will discuss this theorem and related developments, including some recent work by Terence Tao and the speaker which answers the question of how many such lines there must be. Along the way I will have the opportunity to introduce some beautiful and very classical mathematical results.