We present existence and multiplicity theorems for periodic mild solutions to parabolic evolution equations. Their peculiarity is a link with the spectrum of the generator of the semigroup rather than with the spectrum of the linearized periodic BVP for the evolution equation. They provide a positive solution to the open problem risen by Becker [3], they extend some results of Castro and Lazer [5] from scalar to systems of parabolic equations, and they are new even for finite-dimensional ODEs.
