As a new and complex combinatorial optimization problem, the mothership-vehicle collaborative routing problem (MVCRP) has received increasing attention and study from many scholars in recent years. This paper discusses the concept and connotation of the MVCRP, analyzes its main differences from the truck-drone collaborative routing problem, and describes its typical application scenarios in maritime cruise supervision and law enforcement, offshore facility inspection, ship emission detection, and search-and-rescue activities. Based on the mapping relationship formed by the number of motherships and vahicels, this paper provides a systematic overview of the overall research status of the problem and introduces current models and solution methods. Specifically, to further enrich the research, we introduce a new mixed-integer second-order conic programming model for a new variant, where multiple motherships, each equipped with a loaded vehicle, are used to perform oil facility inspection. Finally, the paper provides prospects for future research directions on MVCRP from multiple perspectives including time windows, multiple ports, variable speeds, multiple journeys for one mothership, target with complex shapes, and uncertainties.