In recent years, designing fairness-aware methods has received much attention in various domains, including machine learning, natural language processing, and information retrieval. However, understanding structural bias and inequalities in social networks and designing fairness-aware methods for various research problems in social network analysis (SNA) have not received much attention. In this work, we highlight how the structural bias of social networks impacts the fairness of different SNA methods. We further discuss fairness aspects that should be considered while proposing network structure-based solutions for different SNA problems, such as link prediction, influence maximization, centrality ranking, and community detection. This paper clearly highlights that very few works have considered fairness and bias while proposing solutions; even these works are mainly focused on some research topics, such as link prediction, influence maximization, and PageRank. However, fairness has not yet been addressed for other research topics, such as influence blocking and community detection. We review state-of-the-art for different research topics in SNA, including the considered fairness constraints, their limitations, and our vision. This paper also covers evaluation metrics, available datasets, and synthetic network generating models used in such studies. Finally, we highlight various open research directions that require researchers' attention to bridge the gap between fairness and SNA.