  We survey the different versions of Floer homology that can be associated to
three-manifolds. We also discuss their applications, particularly to questions
about surgery, homology cobordism, and four-manifolds with boundary. We then
describe Floer stable homotopy types, the related Pin(2)-equivariant
Seiberg-Witten Floer homology, and its application to the triangulation
conjecture.
