  General definitions for causal structures on manifolds of dimension d+1>2 are
presented for the topological category and for any differentiable one.
  Locally, these are given as cone structures via local (pointwise)
homeomorphic or diffeomorphic abstraction from the standard null cone variety
in R^{d+1}. Weak and strong local cone (LC) structures refer to the cone itself
or a manifold thickening of the cone respectively.
  After introducing cone (C-)causality, a causal complement with reasonable
duality properties can be defined. The most common causal concepts of
space-times are generalized to the present topological setting. A new notion of
precausality precludes inner boundaries within future/past cones.
  LC-structures, C-causality, a topological causal complement, and precausality
may be useful tools in conformal and background independent formulations of
(algebraic) quantum field theory and quantum gravity.
