  We discuss several topics related to the notion of strong hyperbolicity which
are of interest in general relativity. After introducing the concept and
showing its relevance we provide some covariant definitions of strong
hyperbolicity. We then prove that is a system is strongly hyperbolic with
respect to a given hypersurface, then it is also strongly hyperbolic with
respect to any near by one.
  We then study for how much these hypersurfaces can be deformed and discuss
then causality, namely what is the maximal propagation speed in any given
direction.
  To show that the concept is needed in the area of general relativity we
discuss two results for which the theory of symmetric hyperbolic systems shows
to be insufficient. The first deals with the hyperbolicity analysis of systems
which are second order in space derivatives, they include certain versions of
the ADM and the BSSN families of equations.
  The second result deals with systems having constraints. The question posed
is which hyperbolicity properties, if any, are inherited from the original
evolution system by the subsidiary system satisfied by the constraint
quantities. The answer is that, subject to some condition on the constraints,
if the evolution system is strongly hyperbolic then the subsidiary system is
also strongly hyperbolic and the causality properties of both are identical.
