  The fact that every compact oriented 4-manifold admits spin$^c$ structures
was proved long ago by Hirzebruch and Hopf. However, the usual proof is neither
direct nor transparent. This article gives a new proof using twistor spaces
that is simpler and more geometric. After using these ideas to clarify various
aspects of four-dimensional geometry, we then explain how related ideas can be
used to understand both spin and spin$^c$ structures in any dimension.
