Code generation has emerged as a critical research area at the intersection of Software Engineering (SE) and Artificial Intelligence (AI), attracting significant attention from both academia and industry. Within this broader landscape, Verilog, as a representative hardware description language (HDL), plays a fundamental role in digital circuit design and verification, making its automated generation particularly significant for Electronic Design Automation (EDA). Consequently, recent research has increasingly focused on applying Large Language Models (LLMs) to Verilog code generation, particularly at the Register Transfer Level (RTL), exploring how these AI-driven techniques can be effectively integrated into hardware design workflows. Despite substantial research efforts have explored LLM applications in this domain, a comprehensive survey synthesizing these developments remains absent from the literature. This review fill addresses this gap by providing a systematic literature review of LLM-based methods for Verilog code generation, examining their effectiveness, limitations, and potential for advancing automated hardware design. The review encompasses research work from conferences and journals in the fields of SE, AI, and EDA, encompassing 70 papers published on venues, along with 32 high-quality preprint papers, bringing the total to 102 papers. By answering four key research questions, we aim to (1) identify the LLMs used for Verilog generation, (2) examine the datasets and metrics employed in evaluation, (3) categorize the techniques proposed for Verilog generation, and (4) analyze LLM alignment approaches for Verilog generation. Based on our findings, we have identified a series of limitations of existing studies. Finally, we have outlined a roadmap highlighting potential opportunities for future research endeavors in LLM-assisted hardware design.