Floquet's theorem, and the general qualitative features of the band structure, are described in the one-dimensional case. The discussion includes a little-known unstable state at band edges that is not a Floquet function. This “band edge state” can in certain cases play a role in a thin crystalline layer, analogous to the role of a surface state in a semi-infinite crystal. Some incorrect and misleading statements about Floquet's and Bloch's theorems, which are widespread in expository accounts of the subject, are pointed out.