  The concept of non-Gorenstein involutions on Calabi-Yau threefolds is a
higher dimensional generalization of non-symplectic involutions on $K3$
surfaces. We present some elementary facts about Calabi-Yau threefolds with
non-Gorenstein involutions. We give a classification of the Calabi-Yau
threefolds of Picard rank one with non-Gorenstein involutions whose fixed locus
is not zero-dimensional.
