The utilization of cloud infrastructure and its extensive range of Internet-accessible resources holds significant potential for advancing intelligent transportation and robotics. Over the past two decades, interest in cloud-edge collaborative simultaneous localization and mapping (SLAM) has grown markedly. Consequently, a comprehensive review of current trends in this field is crucial for both novice and experienced researchers. This paper examines robots and automation systems that rely on network-based data or code, particularly in the context of SLAM development. Applying SLAM to mobile robots with limited computing power is essential for achieving autonomous navigation, and cloud-edge collaborative SLAM has emerged as an efficient solution. The review is structured around four key benefits of cloud-edge collaborative SLAM: Assisted Cloud Computing, which provides access to cloud computation and reduces the burden on edge devices; Total Cloud Computing, where the majority of computation is offloaded to the cloud, while edge devices primarily handle sensing and low-cost pre-processing; Data Storage, enabling access to large datasets, such as high-resolution environment maps and extensive training datasets, enhancing overall performance; and Data Transmission, involving cloud-edge communication for efficient data transfer and data association. Additionally, we address the challenges in existing work and the development of asynchronous collaboration and implicit representation transmission, which could mitigate transmission latency in communication-constrained environments. We believe that this review will bridge the gap between SLAM systems and deployed robotic systems, promoting the advancement of cloud-edge collaborative SLAM.