  Let $K$ be a Klein bottle. We show that the infimum of the Willmore energy
among all immersed Klein bottles in Euclidean $n$-space is attained by a smooth
embedded Klein bottle, where $n\geq 4$. There are three distinct regular
homotopy classes of immersed Klein bottles in the Euclidean four-space each one
containing an embedding. One is characterized by the property that it contains
the minimizer just mentioned. For the other two regular homotopy classes we
show that the Willmore energy is bounded from below by $8\pi$. We give a
classification of the minimizers of these two classes. In particular, we prove
the existence of infinitely many distinct embedded Klein bottles in Euclidean
four-space that have Euler normal number $-4$ or $+4$ and Willmore energy
$8\pi$. The surfaces are distinct even when we allow conformal transformations
of the ambient space. As they are all minimizers in their homotopy class they
are Willmore surfaces.
