https://philippelefloch.files.wordpress.com/2015/11/2015-ihp-jpbourguignon.pdf Abstract. Physics and Geometry have a long history in common, but the Theory of General Relativity, and theories it triggered, have been a great source of challenges and inspiration for geometers. It started even before its birth with the motivation coming from the theory of ether in the works of B. Riemann and W. K. Clifford, and the advent of Special Relativity by A. Einstein. The central role Ricci curvature plays in the Theory of General Relativity changed completely the vision geometers have of the importance of this geometric object. Several other instances of geometric notions which were triggered through this interaction will be presented from Conformal Geometry, Riemannian submersions, the space of Riemannian metrics, Killing spinors and topological Lagrangians. These positive interactions culminate of course in the spectacularly deep understanding of the Einstein field equations as a non-linear system of partial differential equation, which will be reported upon at this conference by a number of speakers.