Clustering algorithms are a class of unsupervised machine learning (ML) algorithms that feature ubiquitously in modern data science, and play a key role in many learning-based application pipelines. Recently, research in the ML community has pivoted to analyzing the fairness of learning models, including clustering algorithms. Furthermore, research on fair clustering varies widely depending on the choice of clustering algorithm, fairness definitions employed, and other assumptions made regarding models. Despite this, a comprehensive survey of the field does not exist. In this paper, we seek to bridge this gap by categorizing existing research on fair clustering, and discussing possible avenues for future work. Through this survey, we aim to provide researchers with an organized overview of the field, and motivate new and unexplored lines of research regarding fairness in clustering.