The purpose of this paper, which is largely self-contained though it is a difficult task, is to revisit the mathematical foundations of General Relativity (GR) after one century, in the light of the formal theory of systems of partial differential equations and Lie pseudogroups (D.C. Spencer, 1970) or Algebraic Analysis, namely a mixture of differential geometry and homological algebra (M. Kashiwara, 1970). In particular, we shall justify the claim: GR is not coherent with any one of the above three domains. • Systems: In dimension 4 only, the 9 Bianchi identities that must be satisfied by the 10 components of the Weyl tensor are described by a second order operator and have thus nothing to do with the 20 first order Bianchi identities for the 20 components of the Riemann tensor. This result has been recently confirmed by A. Quadrat (INRIA) using new computer algebra packages. • Groups: The kernel of the canonical projection of the Riemann bundle onto the Weyl bundle, induced by the canonical inclusion of the classical Killing system (Poincaré group) into the confor-mal Killing system (Conformal group), namely the so-called Ricci bundle of symmetric 2-tensors, has only to do with the second order jets of the conformal Killing system. • Modules: The 10 linearized second order Einstein equations are parametrizing the 4 first order Cauchy stress equations but cannot be parametrized themselves. As a byproduct of this negative result, the 4 Cauchy stress equations have nothing to do with the 4 divergence-type equations usually obtained from the Bianchi identities by contraction of indices. These purely mathematical results question the origin and existence of gravitational waves.