V. Alexeev proved in 1994 that the set S of self-intersections of the canonical class of stable surfaces satisfies the descending chain condition, this is, any monotone sequence is increasing. (This set S is a subset of the positive rational numbers.) In particular S has a minimum, and it may have accumulation points. I will discuss what is known about S, certain new theorems on accumulation points, and open questions. This is a joint work with José Ignacio Yáñez.