  We study Fano fourfolds of K3 type with a conic bundle structure. We
construct direct geometrical links between these fourfolds and hyperK\"ahler
varieties. As a result we describe families of nodal surfaces that can be seen
as generalisations of Kummer quartic surfaces. Each of these families actually
arises through two families of Fano fourfolds, whose conic bundle structures
are related by hyperbolic reduction.
