  Hyperbolic conservation laws posed on manifolds arise in many applications to
geophysical flows and general relativity. Recent work by the author and his
collaborators attempts to set the foundations for a study of weak solutions
defined on Riemannian or Lorentzian manifolds and includes an investigation of
the existence and qualitative behavior of solutions. The metric on the manifold
may either be fixed (shallow water equations on the sphere, for instance) or be
one of the unknowns of the theory (Einstein-Euler equations of general
relativity).
