The problem of multi-robot target tracking asks for actively planning the joint motion of robots to track targets. In this paper, we focus on such target tracking problems in adversarial environments, where attacks or failures may deactivate robots' sensors and communications. In contrast to the previous works that consider no attacks or sensing attacks only, we formalize the first robust multi-robot tracking framework that accounts for any fixed numbers of worst-case sensing and communication attacks. To secure against such attacks, we design the first robust planning algorithm, named Robust Active Target Tracking (RATT), which approximates the communication attacks to equivalent sensing attacks and then optimizes against the approximated and original sensing attacks. We show that RATT provides provable suboptimality bounds on the tracking quality for any non-decreasing objective function. Our analysis utilizes the notations of curvature for set functions introduced in combinatorial optimization. In addition, RATT runs in polynomial time and terminates with the same running time as state-of-the-art algorithms for (non-robust) target tracking. Finally, we evaluate RATT with both qualitative and quantitative simulations across various scenarios. In the evaluations, RATT exhibits a tracking quality that is near-optimal and superior to varying non-robust heuristics. We also demonstrate RATT's superiority and robustness against varying attack models (e.g., worst-case and bounded rational attacks).