Over the past decades we have seen robots move from constrained and heavily designed industrial environments out into the unstructured world. This shift drives a need for smaller, safer, and less expensive robots which can collaboratively complete tasks autonomously. For such teams to function, they must be able to reach a shared understanding of their task and environment while accommodating unreliable sensors, and also to recover gracefully from individual failures without requiring centralized coordination. These challenges are the focus of this thesis which lays the groundwork for multi-robot active mapping that is resilient to faulty or malfunctioning sensors. Such algorithms have wide ranging applicability, from persistent monitoring tasks such as those found in agriculture to time critical tasks such as search and rescue. First we present two map representations designed specifically for autonomous information theoretic mapping. For each method, we develop an information theoretic value function which can be used to choose actions to maximize the information gained about the map. We present a principled method for accounting for both information gained by exploring new areas, as well as information gained by further inspection of the existing map to account for sensor uncertainty. To address the vulnerability of local planning methods to local minima, we develop a strategy to maintain a long horizon planning tree over time. Second, we extend a highly distributed approach to resilient consensus for static networks to applications with multi-robot teams. This approach has been largely limited to small static networks or strict formations. First we develop a method which can be used for teams with time-varying range-based communication which is suitable for tasks where robots are not required to spread out in the environment. We then present a method that is well suited to mapping and coverage applications which uses a well known communication structure to guarantee successful resilient consensus. Finally we present examples of how these tools can be used to enable resilient active mapping and coverage for teams of robots with faulty or malfunctioning sensors.