High dimensional geometric data are ubiquitous in science and engineering, and thus processing and analyzing them is a core task in these disciplines. The Computational Geometric Learning project (CG Learning) aims at extending the success story of geometric algorithms with guarantees, as achieved in the CGAL library and the related EU funded research projects, to spaces of high dimensions. This is not a straightforward task. For many problems, no efficient algorithms exist that compute the exact solution in high dimensions. This behavior is commonly called the curse of dimensionality. We plan to address the curse of dimensionality by focusing on inherent structure in the data like sparsity or low intrinsic dimension, and by resorting to fast approximation algorithms. The following two kinds of approximation guarantee are particularly desirable: first, the solution approximates an objective better if more time and memory resources are employed (algorithmic guarantee), and second, the approximation gets better when the data become more dense and/or more accurate (learning theoretic guarantee). To lay the foundation of a new field---computational geometric learning---we will follow an approach integrating both theoretical and practical developments, the latter in the form of the construction of a high quality software library and application software.