Curbs are used as physical markers to delimit roads and to redirect traffic into multiple directions (e.g., islands and roundabouts). Detection of road curbs is a fundamental task for autonomous vehicle navigation in urban environments. Since almost two decades, solutions that use various types of sensors, including vision, Light Detection and Ranging (LiDAR) sensors, among others, have emerged to address the curb detection problem. This survey elaborates on the advances of road curb detection problems, a research field that has grown over the last two decades and continues to be the ground for new theoretical and applied developments. We identify the tasks involved in the road curb detection methods and their applications on autonomous vehicle navigation and advanced driver assistance system (ADAS). Finally, we present an analysis on the similarities and differences of the wide variety of contributions.