Extremely large-scale antenna arrays (ELAA) play a critical role in enabling the functionalities of next generation wireless communication systems. However, as the number of antennas increases, ELAA systems face significant bottlenecks, such as excessive interconnection costs and high computational complexity. Efficient distributed signal processing (SP) algorithms show great promise in overcoming these challenges. In this paper, we provide a comprehensive overview of distributed SP algorithms for ELAA systems, tailored to address these bottlenecks. We start by presenting three representative forms of ELAA systems: single-base station ELAA systems, coordinated distributed antenna systems, and ELAA systems integrated with emerging technologies. For each form, we review the associated distributed SP algorithms in the literature. Additionally, we outline several important future research directions that are essential for improving the performance and practicality of ELAA systems.