Suppose you have n points in the plane, not all on a line. A famous result called the Sylvester-Gallai theorem states that there must be at least one “ordinary line”, by which we mean a line through precisely two points of the set. I will discuss this theorem and then turn to some more recent developments connected with the question of how many ordinary lines there must be. Perhaps unexpectedly, this involves considerations about elliptic curves as well as some results from additive number theory. The talk should be accessible to undergraduates (you won’t need to know what an elliptic curve is).