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As large language models (LLMs) become integral to social and interactive applications, the ability to model, control, and evaluate their personality traits has become a critical area of research. This survey provides a comprehensive and structured overview of the LLM-driven personality scenario. We introduce a functional taxonomy that organizes the field by how personality is modeled (from rule-based methods to model-centric and system-level LLM techniques), across which modalities it is expressed (extending beyond text to vision, speech, and immersive virtual reality), and how it is validated (covering both qualitative and quantitative evaluation paradigms). By contextualizing current advances and systematically analyzing the limitations of existing methods including subjectivity, context dependence, limited multimodal integration, and the lack of standardized evaluation protocols, we identify key research gaps. This survey serves as a guide for future inquiry, paving the way for the development LLMs with more consistent consistent, expressive, and trustworthy personality traits.