  We define Cayley structures as a field of Cayley's ruled cubic surfaces over
a four dimensional manifold and motivate their study by showing their
similarity to indefinite conformal structures and their link to differential
equations. In particular, for Cayley structures an extension of certain notions
defined for indefinite conformal structures in dimension four are introduced,
e.g., half-flatness, existence of a null foliation, ultra-half-flatness, an
associated pair of second order ODEs, and a dispersionless Lax pair. After
solving the equivalence problem we obtain the fundamental invariants, find the
local generality of several classes of Cayley structures and give examples.
