Moduli of curves: new compactifications and flips

I will give an introduction to the Hassett-Keel program,
one of whose goals is to construct a sequence of modular
compactifications of the moduli space of smooth curves by letting the
boundary parameterize worse and worse singular curves. I will
illustrate how the program runs on the examples of the first flip
(work of Hassett and Hyeon) and the second flip (joint work with
Alper, Smyth, and van der Wyck) of the moduli space of stable curves.