Spin^c structures, that are structure that an n-manifold may be endowed with, arise as a sort of generalisation / complexification of Spin structures on n-dimensional manifolds.
They are extremely useful in the study of 3-manifolds because the Heegaard Floer homology, which is one of the most popular recent invariants for 3-manifolds, admits a natural splitting as the sum of the Floer homology groups associated to each Spin^c structure on the manifold.
For the case of 3-manifolds there are several definitions of Spin^c structure, that at a first glance do not really seem connected to each other. The aim of the talk will be to give an idea of what a Spin^c structure is and why the several definitions in the case of 3-manifolds are all the same.