Most of the buildings that exist today were built based on 2D drawings. Building information models that represent design-stage product information have become prevalent in the second decade of the 21st century. Still, it will take many decades before such models become the norm for all existing buildings. In the meantime, the building industry lacks the tools to leverage the benefits of digital information management for construction, operation, and renovation. To this end, this paper reviews the state-of-the-art practice and research for constructing (generating) and maintaining (updating) geometric digital twins. This paper also highlights the key limitations preventing current research from being adopted in practice and derives a new geometry-based object class hierarchy that mainly focuses on the geometric properties of building objects, in contrast to widely used existing object categorisations that are mainly function-oriented. We argue that this new class hierarchy can serve as the main building block for prioritising the automation of the most frequently used object classes for geometric digital twin construction and maintenance. We also draw novel insights into the limitations of current methods and uncover further research directions to tackle these problems. Specifically, we believe that adapting deep learning methods can increase the robustness of object detection and segmentation of various types; involving design intents can achieve a high resolution of model construction and maintenance; using images as a complementary input can help to detect transparent and specular objects; and combining synthetic data for algorithm training can overcome the lack of real labelled datasets.