Videofluoroscopic Swallowing Study (VFSS) is considered the gold standard for diagnosing swallowing disorders, or dysphagia. However, the interpretation of VFSS is susceptible to human bias and subjectivity, resulting in significant inter- and intra-patient variability. In this context, artificial intelligence (AI) has emerged as a potentially valuable tool for physicians. This study reviews state-of-the-art research utilizing AI to analyze VFSS for the assessment of swallowing disorders and to support clinical decision-making. Our comprehensive analysis highlights substantial progress in areas such as pharyngeal phase detection, segmentation and identification of the bolus and hyoid bone, and penetration-aspiration detection. Despite these advancements, an end-to-end automated AI tool for VFSS analysis has yet to be developed. However, there is considerable potential for AI applications in areas like exploring the clinical relevance of segmented or tracked components and expanding the scope to include more upper aerodigestive components in the analysis. Additionally, we discuss the limitations of current research, including the lack of publicly available datasets, the need to address the generalizability of AI models, the integration of cutting-edge AI techniques, and the clinical implications for speech-language pathologists.