Enhancing user engagement through personalization in conversational agents has gained significance, especially with the advent of large language models that generate fluent responses. Personalized dialogue generation, however, is multifaceted and varies in its definition -- ranging from instilling a persona in the agent to capturing users' explicit and implicit cues. This paper seeks to systemically survey the recent landscape of personalized dialogue generation, including the datasets employed, methodologies developed, and evaluation metrics applied. Covering 22 datasets, we highlight benchmark datasets and newer ones enriched with additional features. We further analyze 17 seminal works from top conferences between 2021-2023 and identify five distinct types of problems. We also shed light on recent progress by LLMs in personalized dialogue generation. Our evaluation section offers a comprehensive summary of assessment facets and metrics utilized in these works. In conclusion, we discuss prevailing challenges and envision prospect directions for future research in personalized dialogue generation.