  We present a systematic study of threefolds fibred by K3 surfaces that are
mirror to sextic double planes. There are many parallels between this theory
and the theory of elliptic surfaces. We show that the geometry of such
threefolds is controlled by a pair of invariants, called the generalized
functional and generalized homological invariants, and we derive an explicit
birational model for them, which we call the Weierstrass form. We then describe
how to resolve the singularities of the Weierstrass form to obtain the "minimal
form", which has mild singularities and is unique up to birational maps in
codimension 2. Finally we describe some of the geometric properties of
threefolds in minimal form, including their singular fibres, canonical divisor,
and Betti numbers.
