Compositional generalization is the ability of a model to generalize to complex, previously unseen types of combinations of entities from just having seen the primitives. This type of generalization is particularly relevant to the semantic parsing community for applications such as task-oriented dialogue, text-to-SQL parsing, and information retrieval, as they can harbor infinite complexity. Despite the success of large language models (LLMs) in a wide range of NLP tasks, unlocking perfect compositional generalization still remains one of the few last unsolved frontiers. The past few years has seen a surge of interest in works that explore the limitations of, methods to improve, and evaluation metrics for compositional generalization capabilities of LLMs for semantic parsing tasks. In this work, we present a literature survey geared at synthesizing recent advances in analysis, methods, and evaluation schemes to offer a starting point for both practitioners and researchers in this area.