Point Cloud Registration (PCR) is a basic task in computer vision, mobile robotics, and autonomous driving. PCR primarily faces challenges, including insufficient registration performance in low-overlap scenarios and high computational resource consumption in large-scale point cloud scenarios. Most recent PCR methods are transformer-based. Methods like transformers have quadratic computational complexity $𝒪(n^{2}d)$, leading to rapid increases in computational cost with large-scale point cloud data. To address these problems, an iterative PCR method named Attention and Mamba Based Iterative Registration Network (AMBIR) is proposed, overcoming the shortcomings of the current PCR method on low-overlap and large-scale scenarios. Specifically, an iterative network architecture is introduced that learns overlap experience from prior registration results, thereby enhancing registration performance by leveraging knowledge from the preceding step. Additionally, to convert 3-D point cloud data into linear sequences suitable for the Mamba encoder, the Prior-Informed Co-aligned Serialization is proposed to ensure that points with adjacent indices after serialization are spatial neighbors, thereby improving the efficiency and robustness of the subsequent registration process. Lastly, a Consistency-Aware Mamba Encoder is introduced to leverage its linear computational complexity, making the method more suitable for large-scale point clouds. This method simultaneously overcomes the shortcomings of existing methods, including insufficient registration performance in low-overlap and large-scale point cloud scenarios. It performs well on the 3DMatch dataset, 3DLoMatch low-overlap dataset, and KITTI large-scale scene dataset, demonstrating high practical value.