Abstract  This paper is about a new concept for the description of 3D smooth surfaces: the extremal mesh. In previous works, we have shown how to extract the extremal lines from 3D images, which are the lines where one of the two principal surface curvatures is locally extremal. We have also shown how to extract the extremal points, which are specific points where the two principal curvatures are both extremal. The extremal mesh is the graph of the surface whose vertices are the extremal points and whose edges are the extremal lines: it is invariant with respect to rigid transforms. The good topological properties of this graph are ensured with a new local geometric invariant of 3D surfaces, that we call the Gaussian extremality, and which allows to overcome orientation problems encountered with previous definitions of the extremal lines and points. This paper presents also an algorithm to extract the extremal mesh from 3D images, and experiments with synthetic and real 3D medical images show that this graph can be extremely precise and stable.