Point cloud registration (PCR) involves determining a rigid transformation that aligns one point cloud to another. Despite the plethora of outstanding deep learning (DL)-based registration methods proposed, comprehensive and systematic studies on DL-based PCR techniques are still lacking. In this paper, we present a comprehensive survey and taxonomy of recently proposed PCR methods. Firstly, we conduct a taxonomy of commonly utilized datasets and evaluation metrics. Secondly, we classify the existing research into two main categories: supervised and unsupervised registration, providing insights into the core concepts of various influential PCR models. Finally, we highlight open challenges and potential directions for future research. A curated collection of valuable resources is made available at https://github.com/yxzhang15/PCR.