  We develop causality theory for upper semi-continuous distributions of cones
over manifolds generalizing results from mathematical relativity in two
directions: non-round cones and non-regular differentiability assumptions. We
prove the validity of most results of the regular Lorentzian causality theory
including causal ladder, Fermat's principle, notable singularity theorems in
their causal formulation, Avez-Seifert theorem, characterizations of stable
causality and global hyperbolicity by means of (smooth) time functions. For
instance, we give the first proof for these structures of the equivalence
between stable causality, $K$-causality and existence of a time function. The
result implies that closed cone structures that admit continuous increasing
functions also admit smooth ones. We also study proper cone structures, the
fiber bundle analog of proper cones. For them we obtain most results on domains
of dependence. Moreover, we prove that horismos and Cauchy horizons are
generated by lightlike geodesics, the latter being defined through the
achronality property. Causal geodesics and steep temporal functions are
obtained with a powerful product trick. The paper also contains a study of
Lorentz-Minkowski spaces under very weak regularity conditions. Finally, we
introduce the concepts of stable distance and stable spacetime solving two well
known problems (a) the characterization of Lorentzian manifolds embeddable in
Minkowski spacetime, they turn out to be the stable spacetimes, (b) the proof
that topology, order and distance (with a formula a la Connes) can be
represented by the smooth steep temporal functions. The paper is
self-contained, in fact we do not use any advanced result from mathematical
relativity.
