The group synchronization problem asks to recover group elements, such as 3-D rotations and permutations, on graph nodes from their relative measurements on graph edges. It has critical applications in 3-D reconstruction and many other tasks in data science. It is challenging due to highly corrupted measurements and computational bottlenecks. This talk will demonstrate how to overcome these challenges in a unified framework that applies to all compact groups. The proposed solution is a message passing algorithm that utilizes the cycle consistency of group elements within the graph. The talk will also present several exact recovery guarantees by the proposed method under both adversarial and uniform corruption and demonstrate state-of-the-art accuracy and speed of the proposed method. This new framework may provide a foundation for a next generation of 3-D reconstruction pipeline. The talk will further expand on different developments in 3-D reconstruction of protein molecules via cryo-electron microscopy (cryo-EM).