- In online contingent planning under partial observability an agent decides at each time step on the next action to execute, given its initial knowledge of the world, the actions executed so far, and the observation made. Such agents require some representation of their belief state to determine which actions are valid, or whether the goal has been achieved. Efficient maintenance of a belief state is, given its potential exponential size, a key research challenge in this area. In this paper we develop the theory of regression as a useful tool for belief-state maintenance. We provide a formal description of regression, discussing various alternatives and optimization techniques, and analyze its space and time complexity. In particular, we show that, with some care, the regressed formula will contain variables relevant to the current query only, rather than all variables in the problem description. Consequently, under suitable assumptions, the complexity of regression queries is at most exponential in its contextual width. This parameter is always upper bounded by Bonet and Geffner's width parameter, introduced in their state-of-the-art factored belief tracking (FBT) method. In addition, we show how to obtain a poly-sized circuit representation for the online regression formula even with non-deterministic actions. We provide an empirical comparison of regression with FBT-based belief maintenance, showing the power of regression for online belief tracking. We also suggest caching techniques for regression, and demonstrate their value in reducing runtime in current benchmarks.