Conversational Text-to-SQL systems, which translate natural language dialogues into sequences of SQL queries, aim to democratize data access for non-experts. This survey comprehensively examines the field of multi-turn conversational Text-to-SQL by defining the task and analyzing core challenges, including context handling (such as coreference and ellipsis), semantic flow modeling, and cross-domain generalization. It systematically reviews the evolution of architectural paradigms from early context-aware encoder-decoder models, through sophisticated relation-aware transformers, to recent advancements driven by Large Language Models (LLMs) and agentic frameworks. The article provides a critical quantitative analysis of seminal models against benchmark datasets (SParC and CoSQL), highlighting a clear trend towards more structured and dynamic context representation. Finally, the survey discusses critical unresolved challenges such as error propagation, ambiguity handling, human-in-the-loop refinement, and efficiency of practical deployments, suggesting promising future directions like enhanced agentic systems for complex, multi-step data analysis.