Code generation represents a critical intersection of Software Engineering (SE) and Artificial Intelligence (AI). Within this broader landscape, Verilog, as a representative hardware description language (HDL), is fundamental to Electronic Design Automation (EDA), recent research has increasingly focused on leveraging Large Language Models (LLMs) to automate Verilog code generation, particularly at the Register Transfer Level (RTL) design. Despite growing interest, a comprehensive survey of this domain remains absent. This review addresses this gap by providing a systematic literature review of LLM-based Verilog code generation, analyzing 102 papers (70 published and 32 high-quality preprints) from SE, AI, and EDA venues. We structure our analysis around four key research questions: (1) identifying the LLMs utilized, (2) examining evaluation datasets and metrics, (3) categorizing generation techniques, and (4) analyzing alignment approaches. Furthermore, we synthesize findings to identify critical limitations in current studies regarding effectiveness and integration. Finally, we outline a roadmap highlighting potential opportunities for future research in LLM-assisted hardware design.