  We study threefolds fibred by mirror quartic K3 surfaces. We begin by showing
that any family of such K3 surfaces is completely determined by a map from the
base of the family to the moduli space of mirror quartic K3 surfaces. This is
then used to give a complete explicit description of all Calabi-Yau threefolds
fibred by mirror quartic K3 surfaces. We conclude by studying the properties of
such Calabi-Yau threefolds, including their Hodge numbers and deformation
theory.
