Since the start of the COVID-19 pandemic, social distancing (SD) has played an essential role in controlling and slowing down the spread of the virus in smart cities. To ensure the respect of SD in public areas, visual SD monitoring (VSDM) provides promising opportunities by (i) controlling and analyzing the physical distance between pedestrians in real-time, (ii) detecting SD violations among the crowds, and (iii) tracking and reporting individuals violating SD norms. To the authors’ best knowledge, this paper proposes the first comprehensive survey of VSDM frameworks and identifies their challenges and future perspectives. Typically, we review existing contributions by presenting the background of VSDM, describing evaluation metrics, and discussing SD datasets. Then, VSDM techniques are carefully reviewed after dividing them into two main categories: hand-crafted feature-based and deep-learning-based methods. A significant focus is paid to convolutional neural networks (CNN)-based methodologies as most of the frameworks have used either one-stage, two-stage, or multi-stage CNN models. A comparative study is also conducted to identify their pros and cons. Thereafter, a critical analysis is performed to highlight the issues and impediments that hold back the expansion of VSDM systems. Finally, future directions attracting significant research and development are derived.