A Sylvester-Gallai configuration is a finite set of points C in an affine or projective space such that a line through any two points in C contains a third point of C. It is well-known that any Sylvester-Gallai configuration over R is contained in a line, but this is false over C; the 3-torsion points of an elliptic curve in CP2 provide a counterexample. A conjecture of Serre in 1966, confirmed by Kelly in 1986 using a deep inequality of Hirzebruch, asked whether any Sylvester-Gallai configuration in CP^n was contained in a plane; more recently, Elkies et al. gave an elementary proof. We will discuss these problems and other related work.