Digital Geometry Processing (DGP) started nearly ten years ago on the premise that geometry would soon become the fourth type of digital medium after sounds, images, and video. While recent research efforts have successfully established some theoretical and algorithmic foundations to deal with this very special signal that is geometry, DGP has not resulted in the expected societal and technological impacts that Digital Signal Processing has generated, mostly due to the lack of robustness and genericity of the geometry processing pipeline. We propose a research agenda to harness the full potential of Digital Geometry Processing and make it as robust and impactful as Digital Signal Processing. Specifically, we argue that streamlining the DGP pipeline cannot be achieved by direct adaptation of existing machinery: a new and focused research phase is required to address such fundamental issues as the reconstruction and approximation of complex shapes from heterogeneous data, in order to develop ironclad techniques that are robust to defect-laden inputs and offer strong guarantees on the outputs. Only then can DGP will be ready, as promised, to bring forth a technological revolution.