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This paper surveys Delaunay-based meshing techniques for curved objects, and their application in medical imaging and in computer vision to the extraction of geometric models from segmented images. We show that the so-called Delaunay refinement technique allows to mesh surfaces and volumes bounded by surfaces, with theoretical guarantees on the quality of the approximation, from a geometrical and a topological point of view. Moreover, it offers extensive control over the size and shape of mesh elements, for instance through a (possibly non-uniform) sizing field. We show how this general paradigm can be adapted to produce anisotropic meshes, i.e. meshes elongated along prescribed directions. Lastly, we discuss extensions to higher dimensions, and especially to space-time for producing time-varying 3D models. This is also of interest when input images are transformed into data points in some higher dimensional space as is common practice in machine learning.