In this article, we propose an original solution to the problem of surface reconstruction of large-scale unknown environments, with multiple cooperative robots. As they progress through the 3-D environment, the robots rely on volumetric maps obtained via a TSDF representation to extract discrete incomplete surface elements (ISEs), and a list of candidate viewpoints is generated to cover them. A next-best-view planning approach, which approximately solves a traveling salesman problem (TSP) via greedy allocation, is then used to iteratively assign these viewpoints to the robots. Two multiagent architectures, a centralized one (TSP-Greedy Allocation or TSGA) and a distributed one (dist-TSGA), in which the robots locally compute their maps and share them, are developed and compared. Extensive numerical and real-world experiments with multiple aerial and ground robots in challenging 3-D environments show the flexibility and effectiveness of our surface representation of a volumetric map. The experiments also shed light on the nexus between reconstruction accuracy and surface completeness, and between total distance traveled and execution time.