Maps of LiDAR Simultaneous Localisation and Mapping (SLAM) are often represented as point clouds. They usually take up a huge amount of storage space for large-scale environments, otherwise much structural detail may not be kept. In this paper, a novel paradigm of LiDAR mapping and odometry is designed by leveraging the Continuous and Ultra-compact Representation of LiDAR (CURL) proposed in [1]. Termed CURL-MAP (Mapping and Positioning), the proposed approach can not only reconstruct 3D maps with a continuously varying density but also efficiently reduce map storage space by using CURL’s spherical harmonics implicit encoding. Different from the popular Iterative Closest Point (ICP) based LiDAR odometry techniques, CURL-MAP formulates LiDAR pose estimation as a unique optimisation problem tailored for CURL. Experiment evaluation shows that CURL-MAP achieves state-of-the-art 3D mapping results and competitive LiDAR odometry accuracy. We will release the CURL-MAP codes for the community.