Large language models (LLMs) are proliferating in society, and many occupations are estimated to be impacted by their use and continued development. In fact, in a recent article, Eloundau and colleagues (2023) estimated that survey research is among the fields most vulnerable to this new technology. Survey research has a long-standing history of being a human-powered field, but one that embraces various technologies for the collection, processing, and analysis of various behavioral, political, and social outcomes of interest, among others. At the same time, LLMs bring new technological challenges and prerequisites in order to fully harness their potential within any field. In this paper, we report the results of a systematic literature review based on keyword searches from multiple large-scale databases as well as citation networks that delves deeper into how LLMs are currently being applied within the survey research process. We also discuss new potential use cases for this technology as well as its pitfalls based on examples from existing literature. Results will be synthesized and organized according to the survey research process to include examples of LLM usage for: survey questionnaire development, sampling, data collection, processing, and analysis. Finally, considering survey research has rich experience and history regarding data quality and total error / error sources, we discuss some opportunities and describe future outlooks for survey research to contribute to the continued development and refinement of LLMs.