  We extend the notion of lattice polarization for K3 surfaces to families over
a (not necessarily simply connected) base, in a way that gives control over the
action of monodromy on the algebraic cycles, and discuss the uses of this new
theory in the study of families of K3 surfaces admitting fibrewise symplectic
automorphisms. We then give an application of these ideas to the study of
Calabi-Yau threefolds admitting fibrations by lattice polarized K3 surfaces.
