  This expository article gives a survey of matrix convex sets, a natural
generalization of convex sets to the noncommutative (dimension-free) setting,
with a focus on their extreme points. Mirroring the classical setting, extreme
points play an important role in matrix convexity, and a natural question is,
``are matrix convex sets the (closed) matrix convex hull of their extreme
points?" That is, does a Krein-Milman theorem hold in this setting? This
question requires some care, as there are several notions of extreme points for
matrix convex sets. Three of the most prevalent notions are matrix extreme
points, matrix exposed points, and free extreme points. For each of these types
of extreme points, we examine strengths and shortcomings in terms of a
Krein-Milman theorem. Of particular note is the fact that these extreme points
are all finite-dimensional in nature. As such, a large amount of our discussion
is about free spectrahedra, which are matrix convex sets determined by a linear
matrix inequality.
