Extending a shape-driven map to the interior of the input shape and to the surrounding volume is a difficult problem since it typically relies on the integration of shape-based and volumetric information, together with smoothness conditions, interpolating constraints, preservation of feature values at both a local and global level. This survey discusses the main volumetric approximation schemes for both 3D shapes and d-dimensional data, and provides a unified discussion on the integration of surface-based and volume-based shape information. Then, it describes the application of shape-based and volumetric techniques to shape modeling through volumetric parameterization and polycube splines; feature-driven approximation through kernels and radial basis functions. We also discuss the Hamilton's Ricci flow, which is a powerful tool to compute the conformal shape structure and to design Riemannian metrics of manifolds by prescribed curvatures. We conclude the presentation by discussing applications to shape analysis and medicine.