In conclusion, we have formulated and solved, at a very general level, the problem of tracking targets moving in and out of the field of view of moving sensors. The resulting filters may be applied in numerous settings, such as pursuit-evasion, target enumeration, and situational awareness. When targets move nondeterministically, a combinatorial filter is proposed for the tracking task: We show that the naturally emerging integer linear programming problem is in fact solvable in polynomial time and provide an efficient max-flow based solution for it. For the probabilistic filtering problem in which targets move probabilistically and sensors are not reliable, we give both exact and efficient algorithms that handle the several possible scenarios depending on the number of targets and observations in a system. In solving the more general, probabilistic version of the tracking problem, a clear link is also established between combinatorial filtering and Bayesian filtering methods: The final target distribution is in essence associating the combinatorial solution, a polytope structure, with appropriate probabilities. Viewing it from another angle, the probabilistic shadow information space extends naturally from its nondeterministic counterpart, by merely adding dimensions to record probabilities.