Point clouds are now a popular data representations in 3D vision research. However, due to the limited performance of sensors, the raw data usually suffer from sparsity, sensing noise, and incompleteness. This poses great challenges to downstream point cloud processing tasks. In recent years, deep-learning-based geometric point cloud enhancement methods are gaining tremendous research attention; they use deep neural networks with the aim of achieving dense, clean, and complete point clouds from low-quality raw point clouds. This paper presents a comprehensive survey of deep-learning-based geometric point cloud enhancement methods. It covers three main kinds of geometric point cloud enhancement: (ⅰ) denoising to achieve clean data, (ⅱ) completion to recover unseen data, and (ⅲ) upsampling to obtain dense data. Our survey presents a new taxonomy for recent state-of-the-art methods and systematic experimental results on standard benchmarks. In addition, we share our observations, and suggest future research directions for geometric point cloud enhancement with deep learning.