I will discuss two applications of break divisors (defined in thesecond lecture) to the theory of algebraic curves. The first is a resultof Amini describing the limiting distribution of higher Weierstrass points on an algebraic curve over a non-Archimedean field K. The limiting distribution, called the Zhang measure, has interesting interpretations interms of break divisors, random walks, and electrical networks, and is analogous to the Bergman measure on a Riemann surface. I will also discuss the discretization of the Zhang measure, which is related to the geometric bijections from Lecture 2. The second application I will discuss is the recent work of Tif Shen, from his Yale Ph.D. thesis, relating break divisors to Simpson compactifications of Neron models.