Generic text summarization approaches often fail to address the specific intent and needs of individual users. Recently, scholarly attention has turned to the development of summarization methods that are more closely tailored and controlled to align with specific objectives and user needs. Despite a growing corpus of controllable summarization research, there is no comprehensive survey available that thoroughly explores the diverse controllable attributes employed in this context, delves into the associated challenges, and investigates the existing solutions. In this survey, we formalize the Controllable Text Summarization (CTS) task, categorize controllable attributes according to their shared characteristics and objectives, and present a thorough examination of existing datasets and methods within each category. Moreover, based on our findings, we uncover limitations and research gaps, while also exploring potential solutions and future directions for CTS. We release our detailed analysis of CTS papers at https://github.com/ashokurlana/controllable_text_summarization_survey.