  Motivated by many applications (geophysical flows, general relativity), we
attempt to set the foundations for a study of entropy solutions to nonlinear
hyperbolic conservation laws posed on a (Riemannian or Lorentzian) manifold.
The flux of the conservation laws is viewed as a vector-field on the manifold
and depends on the unknown function as a parameter. We introduce notions of
entropy solutions in the class of bounded measurable functions and in the class
of measure-valued mappings. We establish the well-posedness theory for
conservation laws on a manifold, by generalizing both Kruzkov's and DiPerna's
theories originally developed in the Euclidian setting. The class of {\sl
geometry-compatible} (as we call it) conservation laws is singled out as an
important case of interest, which leads to robust $L^p$ estimates independent
of the geometry of the manifold. On the other hand, general conservation laws
solely enjoy the $L^1$ contraction property and leads to a unique contractive
semi-group of entropy solutions. Our framework allows us to construct entropy
solutions on a manifold via the vanishing diffusion method or the finite volume
method.
