  We study the global theory of linear wave equations for sections of vector
bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of
finite energy sections and show well-posedness of the Cauchy problem in those
spaces. These spaces depend in general on the choice of a time function but it
turns out that certain spaces of finite energy solutions are independent of
this choice and hence invariantly defined.
  We also show existence and uniqueness of solutions for the Goursat problem
where one prescribes initial data on a characteristic partial Cauchy
hypersurface. This extends classical results due to H\"ormander.
