Large Language Models (LLMs) are emerging as promising tools in hardware design and verification, with recent advancements suggesting they could fundamentally reshape conventional practices. In this survey, we analyze over 54 research papers to assess the current role of LLMs in enhancing automation, optimization, and innovation within hardware design and verification workflows. Our review highlights LLM applications across synthesis, simulation, and formal verification, emphasizing their potential to streamline development processes while upholding high standards of accuracy and performance. We identify critical challenges, such as scalability, model interpretability, and the alignment of LLMs with domain-specific languages and methodologies. Furthermore, we discuss open issues, including the necessity for tailored model fine-tuning, integration with existing Electronic Design Automation (EDA) tools, and effective handling of complex data structures typical of hardware projects. This survey not only consolidates existing knowledge but also outlines prospective research directions, underscoring the transformative role LLMs could play in the future of hardware design and verification.