The goal of Facial Kinship Verification (FKV) is to automatically determine whether two individuals have a kin relationship or not from their given facial images or videos. It is an emerging and challenging problem that has attracted increasing attention due to its practical applications. Over the past decade, significant progress has been achieved in this new field. Handcrafted features and deep learning techniques have been widely studied in FKV. The goal of this paper is to conduct a comprehensive review of the problem of FKV. We cover different aspects of the research, including problem definition, challenges, applications, benchmark datasets, a taxonomy of existing methods, and state-of-the-art performance. In retrospect of what has been achieved so far, we identify gaps in current research and discuss potential future research directions.