Large language models (LLMs) are introducing a paradigm shift in molecular discovery by enabling text-guided interaction with chemical spaces through natural language and symbolic notations, with emerging extensions to incorporate multi-modal inputs. To advance this emerging field, this survey provides an up-to-date and forward-looking review of the emerging use of LLMs for two central tasks: molecule generation and molecule optimization. We organize our survey around four fundamental challenges that have emerged as critical evaluation dimensions in recent studies: ensuring validity, enhancing synthesizability, achieving precise property control, and maximizing diversity. Based on this, we systematically analyze how current LLM learning paradigms are applied to tackle each challenge, revealing the distinct capabilities and inherent limitations of each approach. In addition, we include the commonly used datasets and evaluation protocols aligned with these challenges. We conclude by discussing future directions, positioning this survey as a resource for researchers working at the intersection of LLMs and molecular science. A continuously updated reading list is available at https://github.com/REAL-Lab-NU/Awesome-LLM-Centric-Molecular-Discovery.